Photonic-crystal distributed-feedback and distributed bragg-reflector lasers

ABSTRACT

A photonic-crystal distributed-feedback laser includes a laser cavity with a waveguide structure that has a cavity length L c  and is bounded by two mirrors; an active region for producing optical gain upon receiving optical pumping or an input voltage; at least one layer having a periodic two-dimensional grating with modulation of a modal refractive index, the grating being defined on a rectangular lattice with a first period along a first axis of the grating and a second period along a second perpendicular axis of the grating, and wherein the grating produces three diffraction processes having coupling coefficients κ 1 ′, κ 2 ′, κ 3 ′; and a lateral gain area contained within a second area patterned with the grating that has substantially a shape of a gain stripe with a width W, with the gain stripe tilted at a first tilt angle relative to the two mirrors. The rectangular lattice of the grating is tilted at a second tilt angle substantially the same as the first tilt angle with respect to the gain stripe, and the ratio of the first and second grating periods is equal to the tangent of the first tilt angle, with the first tilt angle being between about 16° and about 23°. The hexagonal lattice does not need to be tilted with respect to the two mirrors. The laser&#39;s output emerges along the normal to a facet irrespective of the operating laser wavelength, facilitating coupling the laser light into a fiber or other optical system while avoiding beam steering. The two-dimensional nature of the feedback in the laser provides for varying the wavelength through angle tuning. Wavelength tuning by changing the propagation direction (propagation angle) permits a straightforward selection of different wavelengths from photonic crystal devices monolithically fabricated on a single wafer. The fabrication procedure is straightforward since no ridges need to be defined. The single-mode spectral purity of the rectangular-lattice PCDFB is robust, owing to the near absence of side modes, and exhibits good beam quality.

The present invention claims priority from U.S. Provisional ApplicationNo. 60/362,984, filed Mar. 7, 2002, entitled “Photonic-CrystalDistributed-Feedback and Distributed Bragg-Reflector Laser”,incorporated herein by reference.

TECHNICAL FIELD

The present invention relates to photonic crystal based lasers. Moreparticularly, the invention relates to photonic crystal distributedfeedback lasers.

BACKGROUND ART

An ideal semiconductor laser that would emit high power into a singlespectral mode with diffraction-limited output profile is of greatinterest in a number of applications, including spectroscopy orwavelength multiplexing in telecommunications. However, known laserconfigurations have been unable to provide the desireddiffraction-limited output profile without sacrificing power output, andvice versa. For example, in one approach, a distributed-feedback (DFB)configuration using a one-dimensional (1D) diffraction grating parallelto the laser facets provides high spectral purity when the waveguide issufficiently narrow, e.g. on the order of 2–5 μm for lasers emitting at0.8–1.55 μm wavelengths, thereby suppressing higher-order lateral modeswith respect to the fundamental mode. Scaling up the stripe widthprovides increased power but has the drawback of producing a loss ofphase coherence across the DFB laser stripe, primarily due to theself-modulation of the refractive index in the active region bynon-uniformly distributed carriers. The laser output spectrum isundesirably broadened and limited by the width of the gain spectrum,producing a rapidly diverging, often double-lobed, far-field pattern.

The use of two-dimensional (2D) gratings has been studied employingcoupled-mode theory, e.g. “Proposed cross grating single-mode DFBlaser”, M. Toda, IEEE J. Quantum Electron. (1992) and “Two-dimensionalrectangular lattice distributed feedback lasers: a coupled-mode analysisof TE guided modes”, H. Han and J. J. Coleman, IEEE J. Quantum Electron.(1995). However, these approaches were directed to superimposing 1Dgratings in lieu of using actual 2D gratings which allows only twodiffraction processes. Also, realistic device geometries and thecritical role played by the linewidth enhancement factor (LEF) were notconsidered.

Another approach, described in U.S. Pat. No. 3,970,959 to Wang et al.,is directed to utilizing a DFB laser with a 2D grating to produceperiodic perturbations of an acoustic wave by the photo-elastic effect.The approach, however, merely involved varying the refractive indexwithout disclosing device parameters or a 2D lattice structure. Otherpublications, such as “Coherent two-dimensional lasing action insurface-emitting laser with triangular-lattice photonic crystalstructure”, M. Imada et al., Appl. Phys. Lett. (1999), while disclosingexperimental demonstrations of the 2D distributed feedback (DFB) lasers,are limited to surface-emitting schemes.

Yet another approach disclosed in OSA Topical Meeting on AdvancedSemiconductor Lasers and Applications, Paper AWA6, Kalluri et al. (1999)is directed to an edge-emitting device with a 2D photonic crystalgrating. However, the selected geometry, with the facets being tiltedrelative to the grating to achieve emission normal to the facet,produces an output consisting of two beams emerging at large angles tonormal, not a single near-diffraction-limited beam normal to the facet.

The α-DFB laser, disclosed in “Theory of grating-confined broad-arealasers”, J. Lang, K. Dzurko, A. A. Hardy, S. DeMars, A. Schoenfelder,and D. F. Welch, IEEE J. Quantum Electron. (1998), is directed to aconfiguration in which both the diffraction grating and the gain stripeare tilted with respect to the laser facets. However, both the cavityfacets and the grating are necessary to produce optical feedback andlaser oscillation is established when the beam approaches the facetnearly at normal incidence, since only then is the optical wave stronglyreflected back into the cavity along the same path for feedback. It isprimarily this reflection of only a narrow angular cone at the facetsthat leads to diffraction-limited output beams for much wider pumpstripes than normally attainable in the Fabry-Perot geometry. A problemwith this is that none are robust enough to maintain a single mode underall conditions of interest.

In the approach disclosed in “Single-mode spectral output observed undercw pumping in shorter-wavelength α-DFB lasers”, A. M. Sarangan et al,IEEE J. Quantum Electron. (1995) and “Far-field characteristics ofmid-infrared angled-grating distributed-feedback lasers”, I. Vurgaftmanet al., J. Appl. Phys., (2000), the near-IR laser line broadened inpulsed mode while mid-IR devices exhibited little spectral narrowing.This indicates that the spectral selectivity of α-DFB lasers is ingeneral considerably lower than in other DFB lasers. A small deviationin the angle of orientation translates into a large (on the scale of thegain bandwidth) shift in the emission wavelength. Angled-grating (α-DFB)device configurations have been disclosed, for example, in U.S. Pat. No.5,337,328 to Lang et al. and in U.S. Pat. No. 6,122,299 to DeMars et al.

U.S. Pat. No. 6,052,213 to Burt et al. was concerned with fabricating adiffraction grating in a semiconductor wafer, which can in principlereplace a bulk optical grating component. A problem with this patent isthat the 2D pattern is limited to no more than 10 rows, preferably 1–3rows and the grating is used simply to disperse the incoming light atgrazing incidence to the sample surface.

“Two-dimensional photonic band-gap defect mode laser”, O. Painter, R. K.Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim,Science, vol. 284, pp. 1819–1821 (1999), discloses photonic bandgaplasers and optical components employing 2D photonic crystals as a meansof suppressing spontaneous emission into unwanted optical modes orcreating a microcavity or waveguide. The disclosed device seeks toachieve a large photonic bandgap, usually at as wide a range of anglesas possible. High output power can be combined with good beam quality inthe tapered-laser design of “1.9-W quasi-CW from anear-diffraction-limited 1.55-μm InGaAsP—InP tapered laser”, S. H. Cho,S. Fox, F. G. Johnson, V. Vusiricala, D. Stone, and M. Dagenais, IEEEPhoton. Technol. Lett. (1998). The tapered laser consists of twodistinct sections of the optical waveguide. The first section is asingle-mode ridge waveguide, and the second is a funnel-shapedgain-guided region. A combination of high-reflectivity (HR) andantireflection (AR) coatings insures that nearly all of the laser lightemerges from the tapered section end. However, the output power oftapered lasers is constrained by the maximum aperture that can producediffraction-limited output for a given injected current density and thedesign becomes less attractive for materials with substantialcarrier-induced refractive-index fluctuations.

Single-mode operation can also be obtained with a narrow-stripedistributed Bragg reflector (DBR) laser, in which the distributedfeedback is confined to mirror-like gratings at one or both ends of theoptical cavity. While the grating does not extend into the centralregion, the feedback from the DBR mirrors is wavelength-selective unlikethat in Fabry-Perot lasers. DBR lasers face the same problems as DFBemitters with regard to deterioration of the beam quality and side modesuppression as the stripe is broadened. Widely tunable lasers are known,e.g. as disclosed in “Optimization of the carrier-inducedeffective-index change in InGaAsP Waveguides—Application to tunableBragg filters”, J. P. Weber, IEEE J. Quantum Electron. (1994). However,the spectral range is limited by the magnitude of the interband andintervalence absorption contributions to the refractive index, since theplasma shift is relatively small at the telecommunications wavelength of1.55 μm. Sampled-grating DBR lasers, in which the tuning range hasreached 72 nm are also known, as disclosed in “Theory, design, andperformance of extended tuning range semiconductor lasers with sampledgratings”, V. Jayaraman et al., IEEE J. Quantum Electron (1993). Adrawback with this type of laser is that very narrow ridge waveguidesare required in order to assure lateral coherence of the laser beam.

None of these approaches provide guidance for identifying the optimizedparameters (aspect ratio, etch depth, grating feature size etc.) forstructures with 2D gratings. They also do not consider including allthree relevant diffraction processes for the rectangular lattice, or,apart from Kalluri et al., tilting the grating.

DISCLOSURE OF THE INVENTION

According to the invention, a wavelength-tunable multi-section laser forproducing an output normal to one of two opposing facets includes alaser cavity bounded by i) a first edge and a second opposing edge, andii) a first such facet and a second, opposing facet. The laser cavityincludes: a waveguide structure; a central section, for producing gainupon receiving a first input voltage, that is defined by a firstboundary and a second opposing boundary, and that includes an activeregion and has a first modal refractive index; a first grating withperiod Λ₁, extending between the first and second facets, which ispositioned perpendicular to the first and second boundaries and isinclined at an angle θ₂ relative to the facets; and a first tuningsection, located between the first boundary and the first facet, havinga first tuning electrode and a second modal refractive index. The laserfurther includes a second input voltage for varying the second modalrefractive index to tune the wavelength of the output.

Also according to the invention, a photonic-crystal distributed-feedbacklaser includes: a laser cavity with a waveguide structure that has acavity length L_(c) and is bounded by two mirrors; an active region forproducing optical gain upon receiving optical pumping or an inputvoltage; at least one layer having a periodic two-dimensional gratingwith modulation of a modal refractive index, the grating being definedon a rectangular lattice with a first period along a first axis of thegrating and a second period along a second perpendicular axis of thegrating, and wherein the grating produces three diffraction processeshaving coupling coefficients κ₁′, κ₂′, κ₃′; and a lateral gain areacontained within a second area patterned with the grating that hassubstantially a shape of a gain stripe with a width W, with the gainstripe tilted at a first tilt angle relative to the two mirrors. Therectangular lattice of the grating is tilted at a second tilt anglesubstantially the same as the first tilt angle with respect to the twomirrors, and the ratio of the first and second grating periods is equalto the tangent of the first tilt angle, with the first tilt angle beingbetween about 16° and about 23°.

Also according to the invention, a photonic-crystal distributed-feedbacklaser includes: a laser cavity with a waveguide structure that has acavity length L_(c) and is bounded by two mirrors; an active region forproducing optical gain upon receiving optical pumping or an inputvoltage; at least one layer having a periodic two-dimensional gratingwith modulation of a modal refractive index, the grating being definedon a hexagonal lattice with a first period along a first axis of thegrating and a second period along a second perpendicular axis of thegrating, and wherein the grating produces three diffraction processeshaving coupling coefficients κ₁′, κ₂′, κ₃′; and a lateral gain areacontained within a second area patterned with the grating that hassubstantially a shape of a gain stripe with a width W, with the gainstripe tilted at a first tilt angle relative to the two mirrors. Therectangular lattice of the grating is tilted at a second tilt anglesubstantially the same as the first tilt angle with respect to the twomirrors, and the ratio of the first and second grating periods is equalto the tangent of the first tilt angle, with the first tilt angle beingbetween about 16° and about 23°.

Also according to the invention, a photonic-crystal distributed-feedbacklaser includes a laser cavity in the shape of a microdisk with adiameter D of at least about 100 μm; an active region for producingoptical gain upon receiving optical pumping or an input voltage; atleast one layer having a periodic square-lattice grating with modulationof a modal refractive index, wherein the grating produces threediffraction processes having coupling coefficients κ₁ and κ₂ satisfyingthe constraints: 0.5≦|κ₁|D≦2.0; and 0.5≦|κ₂|D≦2.0; and a lateral gainarea bounded by the microdisk cavity.

Also according to the invention, a photonic-crystal distributed-feedbacklaser includes a laser cavity bounded by at least one mirror; an activeregion for producing optical gain upon receiving optical pumping or aninput voltage; at least one layer including a periodic two-dimensionalgrating with modulation of a modal refractive index; a lateral gainarea; and a resonant lasing wavelength that can be tuned by varying anangle of propagation with respect to a Brillouin zone boundary of anoblique lattice so that said wavelength tuning is inversely proportionalto the cosine of any differential change in said angle of propagation.

Also according to the invention, a photonic-crystal distributed-feedbacklaser includes: a laser cavity bounded by two mirrors; an active regionfor producing optical gain upon receiving optical pumping or an inputvoltage; at least one layer including a periodic line grating withmodulation of a modal refractive index; a lateral gain area; and atleast one layer wherein a second line grating is formed by injectingcarriers in a periodic pattern using an optical pump spatially modulatedalong a direction perpendicular to said line grating.

The invention has several advantages over conventional devices. Thelaser output emerges along the normal to a facet irrespective of theoperating laser wavelength, thereby facilitating coupling the laserlight into a fiber or other optical system while avoiding beam steering.The PCDFB scheme is robust, provides good side-mode suppression ratio,and does not necessitate that the Γg_(th)α product be low.

Furthermore, the two-dimensional nature of the feedback in these lasersprovides for varying the wavelength through angle tuning, since in aphotonic crystal the resonance wavelength is not fixed by the period asin 1D DFB and DBR lasers but varies as a function of the propagationdirection. Wavelength tuning by changing the propagation direction(propagation angle) permits a straightforward selection of differentwavelengths from photonic crystal devices monolithically fabricated on asingle wafer. The fabrication procedure is straightforward since noridges need to be defined.

The PCDFB laser of the invention yields a nearly diffraction-limitedoutput. The single-mode spectral purity of the rectangular-lattice PCDFBis robust, owing to strong side-mode suppression, and exhibits good beamquality.

The present invention also provides a substantial extension of theconditions under which single-mode operation can be sustained in thepresence of LEF-driven index fluctuations. The PCDFB has an extensivesingle-mode region, and single-mode output powers may be up to an orderof magnitude higher in comparison with conventional lasers. Thehexagonal-lattice PCDFB can produce a single-mode output at stripewidths of up to approximately 1.5 mm, and yield single-mode powershigher than α-DFB devices by a factor of approximately 2.5. The PCDBRallows optical coherence to be maintained over a much wider gain stripe,and therefore enables higher powers to be emitted into a single mode.

Many laser configurations are possible, for example, separable-gratingschemes and schemes based on superstructure gratings or chirpedgratings. In these cases, the maximum output power of thewavelength-tunable single mode is substantially enhanced when inventivePCDBR mirrors are employed because coherence can be maintained for amuch wider stripe. Electrically-pumped PCDFB devices will operate up tothe maximum modulation frequencies characteristic of state-of-the-artconventional DFB devices, i.e., in the GHz range.

Additional features and advantages of the present invention will be setforth in, or be apparent from, the detailed description of preferredembodiments which follows.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic representation of the rectangular-lattice PCDFBlaser with circular grating features and a tilted gain stripe accordingto the present invention.

FIG. 2 shows a contour plot of the single mode operation regions as afunction of pump stripe and Γg_(th)α product for a 1D DFB laser (black),an α-DFB laser (gray), and an optimized PCDFB Laser (light gray).

FIG. 3 shows a schematic representation of the hexagonal-lattice PCDFBlaser having circular grating features and three propagation axes, withP₁ being the output direction.

FIG. 4 shows a schematic representation of a single-facethexagonal-lattice PCDFB laser according to the present invention.

FIG. 5 shows a schematic representation of a laser with a hexagonal (ortilted rectangular) grating and a curved facet used for beam steeringaccording to the present invention, wherein the output angle varies withthe pump stripe position.

FIG. 6 shows a schematic representation of a laser with a convergingoutput according to the present invention, wherein the radius ofcurvature is comparable to the stripe width and the laser is tuned tofast-axis angle for near-circular output.

FIG. 7 shows a schematic representation of a microdisk laser accordingto the present invention with a higher efficiency for smaller and largerdiameters than conventional microdisks.

FIG. 8 shows a schematic representation of a multi-sectionwavelength-tunable PCDFB laser according to the present invention.

FIG. 9 shows a schematic representation of an oblique-lattice PCDFBstructure according to the present invention.

FIG. 10 shows a schematic representation of a multi-sectionwavelength-tunable PCDFB laser according to the present inventionwherein each laser in the array employs the same photonic crystalfeedback to produce multiple angles and emission wavelengths.

FIG. 11 shows a schematic representation of a multi-sectionwavelength-tunable PCDFB laser according to the present inventionwherein each laser in the array employs the same PCDBR feedback toproduce multiple angles and emission wavelengths.

FIG. 12 shows a schematic representation of a tunable 2D gratingconsisting of two superimposed line gratings, one of which is etched andthe other defined by a modulated optical pump beam.

FIG. 13 shows a plot of the etendue (in units of the diffraction limit)as a function of stripe width for Fabry-Perot (FP), α-DFB, and PCDFBlasers pumped by a top-hat profile at 14 times threshold. For all threelasers the wavelength is 4.6 μm, the cavity length is 2 mm, and theinternal loss is 20 cm⁻¹. For the (α-DFB and PCDFB lasers, θ=20° andκ₂=200 cm⁻¹, while and κ₁=κ₃=0 for the α-DFB laser and κ₁=κ₃=10 cm⁻¹ forthe PCDFB laser.

FIG. 14 shows a plot of the etendue (in units of the diffraction limit)as a function of stripe width for Fabry-Perot (FP), α-DFB, and PCDFBlasers with the same properties as those of FIG. 11 except that LEF=2and κ₂=50 cm⁻¹ for the α-DFB and PCDFB lasers.

FIG. 15 shows a plot of the etendue (in units of the diffraction limit)as a function of stripe width for Fabry-Perot (FP), α-DFB, and PCDFBlasers with the same properties as those of FIG. 11 except that LEF=0.2and κ₂=50 cm⁻¹ for the α-DFB and PCDFB lasers.

FIG. 16 shows a plot of the emission spectrum of the PCDFB (solid curve)laser and the α-DFB (dashed line) laser, with a stripe width of 100 μm,a Γg_(th)α product of 40 μm, and an emission wavelength of λ=4.6 μm.

BEST MODE FOR CARRYING OUT THE INVENTION

With reference to the drawings, FIG. 1 shows a two-dimensional photoniccrystal (PCDFB) laser geometry (10) comprising a lattice symmetry thatis rectangular, along with an angle θ (12) that specifies an aspectratio of the two lattice periods, Λ₁ (14) and Λ₂ (16). The periods, Λ₁(14) and Λ₂ (16) are representative of the spacing between the gratingfeatures (24) along the x-axis (Λ₁) and the z-axis (Λ₂). Although theinvention is described referencing a rectangular geometry and TE- andTM-polarized light equations specialized to the case of a rectangularlattice, it should be understood that other lattice symmetries such ashexagonal and triangular lattices utilizing couplings between the sixequivalent Γ-X directions in the reciprocal space are also included. Thetype of geometry selected depends on the properties of the gain mediumand the output desired from the laser.

In the PCDFB configuration of FIG. 1, there are two propagationdirections, P₁ (30) and P₂ (34). The two cleaved edges (18 and 20), orfacets are aligned with one (30) of the two coupled propagationdirections (P₁ in FIG. 1) so that the laser output beam (22) emerges ina direction nearly normal to the facet to facilitate collection andcollimation. However, other alignments are possible depending on thedesired laser beam alignment, while the facets may be uncoated or coated(either AR-coated, HR-coated, or one of each). For the purpose ofdescribing the design method of the present invention, uncoated facetsare described herein.

The rectangular lattice shown in FIG. 1 has circular shallow gratingfeatures (24) of radius r that produce modulation of the refractiveindex. Modulation can also be attained using grating features with ashape other than circular (such as square, rectangle, ellipse, or somemore arbitrary shape). Also, although the grating features (24) shown inFIG. 1 are assumed to be etched holes, other well-known means toperiodically modulate the refractive index (such as etched pillars, thedeposition of a material with different optical constants, etc.) may beused equally well to provide these features.

For second-order coupling, the required index modulation must be nearlyquadrupled in order to obtain coupling coefficients of the samemagnitude. And, although the periods (14 and 16) in the second-ordergrating is roughly twice that of the first-order grating, the optimumgrating feature (24) size is only slightly larger. Also, second-ordergratings have the advantage of more relaxed fabrication tolerances.However, this must be weighed against the disadvantage of considerablylarger Δn and the possibility of losses due to diffraction intosurface-emitting modes. Whether a first-order or second-order grating ispreferable therefore depends on practical details of the laser, thelaser fabrication, and also on its intended application. For thepurposes of the following discussion, a first-order grating is used.

The tilted-stripe geometry of the gain stripe (26) shown in FIG. 1 hasits long (z) axis aligned with the short side of the primitive unit cellin real space and its short (x) axis aligned with the long side of thesame unit cell. This orientation is required to obtain the maximumefficiency of the laser device, although small misalignments of the gainstripe (26) with respect to the photonic lattice do not appreciablyaffect other laser characteristics. Since local index fluctuationsinduced by the variation of the carrier density through the LEF aresuppressed by the lossy regions (38 and 39) surrounding the gain stripe(26), this configuration allows Fabry-Perot-like modes to propagatethrough the laser cavity without interacting with the very narrowgrating resonance. The weakly gain-guided structure of FIG. 1 can bemade strongly gain-guided by preventing gain due to “secondary” pumpingof the regions (38) outside of the gain stripe by the emitted photons.This is accomplished by fabricating the laser with ion-bombardingregions that are not subject to appreciable “primary” pumping, orotherwise preventing gain in the regions outside the gain stripe (38 and39) while maintaining the grating periodicity. However, it isdetrimental to the laser performance if strong gain-guiding is inducedby etching a mesa or otherwise interrupting the periodic modulation ofthe refractive index, since said periodic modulation extending beyondthe gain stripe (26) is required to maximize the optical coherenceacross broad stripes. The present invention includes laser devices inwhich gain is created by either optical pumping or electrical injection.

The design method according to the present invention begins with thewave equation: $\begin{matrix}{{{\frac{1}{n_{m}^{2}(r)}{\nabla^{2}E_{y}}} - {\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}E_{y}}} = 0} & (1)\end{matrix}$where E is the TE-polarized electric field, n_(m) is the modal index ateach point r in the laser plane, the strong confining effect of thewaveguide along the growth direction is included, and all vectors andassociated operators here and below represent 2D (i.e., in the x-zplane) quantities. The dispersion (apart from the variation of theoptical gain) is not considered, although it can be included by usingthe group index in place of the modal index in appropriate places. Formost lasers, the group index differs from the modal index by arelatively weak correction of no more than a few percent.

An operating wavelength is selected to achieve the optimum beam qualityor cohesion and power spectrum. The wavelength, λ_(c)=2πc/ω₀, isselected to match the peak of the gain spectrum of the laser activeregion, since this is generally the most advantageous condition.Optimally de-tuning may be carried out to reduce the value of thelinewidth enhancement factor (LEF), if so desired.

A waveguide structure, which typically includes an active layer, top andbottom cladding layers, and a separate confinement layer, must beselected. The selection of the active region (or cavity) of the laser isbased upon a consideration of the LEF and the internal loss factor. Adielectric constant and a complex carrier contribution C (which is afunction of the carrier density N) are associated with the activeregion. The dielectric constant can be expanded in terms of plane waves,and the result can be conveniently written down as follows:$\begin{matrix}{\frac{n_{0}^{2}}{n_{m}^{2}(r)} = {{\sum\limits_{\; G}^{\;}\;{{\hat{\kappa}(G)}{\mathbb{e}}^{{\mathbb{i}}\;{G \cdot r}}}} + {{in}_{0}\frac{c}{\omega_{0}}{\hat{G}\left( {r,\mspace{11mu} N} \right)}}}} & (2)\end{matrix}$where n₀ is the average modal index in the absence of carriers, n_(m) isthe modal index at each r in the laser plane, G represents reciprocallattice vectors, and {circumflex over (κ)} represents the 2D plane-waveexpansion of the normalized dielectric constant. The carriercontribution (assumed to be much smaller than the real part of the modalindex) may be written out explicitly as: Ĝ=Γg(1−iα)−α_(i), where Γ isthe confinement factor obtained from the 1D eigenvalue calculation,g(r,N) is the material gain, α is the LEF, and α_(i) is the internalloss. After factoring out the fast spatial variation in the electricfield introduced by the 2D periodic pattern, a slow-varying functiona_(k)(r, t), which is governed by the active carrier distribution andboundary conditions, is left: $\begin{matrix}{{E\left( {r,\mspace{11mu} t} \right)} = {\sum\limits_{\; G}^{\;}\;{{a_{k + G}\left( {r,\mspace{11mu} t} \right)}{\mathbb{e}}^{{\mathbb{i}}{\lbrack{{{({k + G})} \cdot r} - {\omega\; t}}\rbrack}}}}} & (3)\end{matrix}$where E is the electric field, and k is the wave vector.

Substituting these elements into the wave equation, the followingequations for the TE-polarized optical field $\begin{matrix}{\left. {{{{{\sum\limits_{\; G}^{\;}\;{{{\hat{\kappa}\left( {G - G^{\prime}} \right)}\left\lbrack \quad \right.}{\left( {k + G} \right) \cdot \left( {k + G^{\prime}} \right)}a_{k + G^{\prime}}}} - {{i\left( {{2k} + G + G^{\prime}} \right)} \cdot {\overset{\_}{\nabla}a_{k + G^{\prime}}}} -}\quad}\quad}{\nabla^{2}a_{k + G^{\prime}}}} \right\rbrack = {\left\lbrack {{\frac{\omega_{0}^{2}}{c^{2}}n_{0}^{2}} - {i\frac{\omega_{0}}{c}n_{0}\hat{G}} + {i\frac{2\omega_{0}}{c^{2}}n_{0}^{2}\frac{\partial\;}{\partial t}}} \right\rbrack a_{k + G}}} & (4)\end{matrix}$and for TM-polarized light $\begin{matrix}{\left. {{\sum\limits_{\; G^{\prime}}^{\;}\;{{\hat{\kappa}\left( {G - G^{\prime}} \right)}\left\lfloor \left| {k + G^{\prime}} \right|^{2}\quad \right.a_{k + G^{\prime}}}} - {{i\left( {{2k} + G + G^{\prime}} \right)} \cdot {\overset{->}{\nabla}a_{k + G^{\prime}}}} - {\nabla^{2}a_{k + G^{\prime}}}} \right\rfloor = {\left\lbrack {{\frac{\omega_{0}^{2}}{c^{2}}n_{0}^{2}} - {i\frac{\omega_{0}}{c}n_{0}\hat{G}} + {i\frac{2\omega_{0}}{c^{2}}n_{0}^{2}\frac{\partial\;}{\partial t}}} \right\rbrack a_{k + G}}} & (5)\end{matrix}$can be obtained. In principle, these equations can be applied to anyphotonic-crystal device. The only assumption made is the existence of areasonably slow variation (in time and space) of the envelope function.The key simplification in the subsequent analysis is that the refractiveindex perturbation introduced by the grating features (24) is assumed tobe small enough that the expansion of the equations for TE- andTM-polarized light can be accurately carried out using a very limitednumber of reciprocal lattice vectors G. The accuracy of this assumptionis excellent as long as Δn/n₀ is no larger than a few percent. It isfurther assumed that the unperturbed modal index n₀ is constant,although lateral and longitudinal variations are easily incorporated. Aninvariant grating period (14 and 16) and shape (24) is also assumed,although the etch depth can be varied. The cavity length (28) isspecified by L_(c).

The index perturbation characteristic of the PCDFB lasers in thegeometry of FIG. 1 is small, so only components corresponding to thefollowing four vectors are considered: P₁={l/2,m/2} (30),−P₁={−l/2,−m/2} (32), P₂={−l/2,m/2} (34), and −P₂={l/2,−m/2} (36), wherethe notation {l,m} defines the order of diffractive coupling in terms ofprimitive reciprocal lattice vectors. That is, {l,m}=lb₁+mb₂, b₁=2β(sinθ,0), and b₂=2β(0,cos θ), where β=π/(Λ₂ cos θ) is the diagonal distancefrom the center to the corner of the 1^(st) Brillouin zone. Expandingthe propagation vectors in terms of reciprocal lattice vectors:±P_(1,2)=(k+G_(±1,±2)), it is straightforward to obtain: k={1/2,1/2},G₁={0,0}, G⁻¹={−1,−1}, G₂={−1,0}, and G⁻²={0,−1} for the {1,1} couplingorder; k=0,0}, G₁={1,1}, G⁻¹={−1,−1}, G₂={−1,1}, and G⁻²={1,−1} for the{2,2} coupling order; k={1/2,1/2}, G₁={1,0}, G⁻¹={−2,−1}, G₂={−2,0}, andG⁻²={1,−1} for the {3,1} coupling order, etc. The four propagationdirections have non-zero coupling if the corresponding vctors: (1) haveequal amplitude |k+G_(i)|=|k+G_(j)| and (2) are connected by somereciprocal lattice vector G_(j)−G_(i). For example, coupling orders withl+m equal to an odd integer have only two equivalent directions becauseof condition (1). Similarly, any four equivalent directions thatcorrespond to non-integer l, m do not fulfill condition (2). Theimplication is that the coupled propagation directions for anynon-trivial rectangular PCDFB lattice point along diagonals of theWigner-Seitz cell, i.e., l>0, m>0, and l+m=2, 4, 6, . . . .

When coupling orders with l=m and substituting the vectors ±P_(1,2) intoEq. (4), the following equations are obtained for the TE polarization:$\begin{matrix}{{{\left( {{k_{0}^{2}n_{0}^{2}} - {{ik}_{0}n_{0}\hat{G}} - {\beta^{2}m^{2}}} \right)a_{1}} + {2{ik}_{0}n_{0}\frac{n_{0}}{c}\frac{\partial a_{1}}{\partial t}} + {2i\;\beta\;{m\left( {{\cos\;\theta\frac{\partial a_{1}}{\partial z}} + {\sin\;\theta\frac{\partial a_{1}}{\partial x}}} \right)}} + \frac{\partial^{2}a_{1}}{\partial x^{2}} + \frac{\partial^{2}a_{1}}{\partial z^{2}}} = {2{m\left\lbrack {{{- \kappa_{1}}\beta\;{\overset{\_}{a}}_{1}} + {\kappa_{2}{{\beta cos}\left( {2\theta} \right)}a_{2}} - {\kappa_{3}{{\beta cos}\left( {2\theta} \right)}{\overset{\_}{a}}_{2}}} \right\rbrack}}} & (6) \\{{{\left( {{k_{0}^{2}n_{0}^{2}} - {{ik}_{0}n_{0}\hat{G}} - {\beta^{2}m^{2}}} \right)a_{2}} + {2{ik}_{0}n_{0}\frac{n_{0}}{c}\frac{\partial a_{2}}{\partial t}} + {2i\;\beta\;{m\left( {{\cos\;\theta\frac{\partial a_{2}}{\partial z}} - {\sin\;\theta\frac{\partial a_{2}}{\partial x}}} \right)}} + \frac{\partial^{2}a_{2}}{\partial x^{2}} + \frac{\partial^{2}a_{2}}{\partial z^{2}}} = {2{m\left\lbrack {{{- \kappa_{1}}\beta\;{\overset{\_}{a}}_{2}} + {\kappa_{2}{{\beta cos}\left( {2\theta} \right)}a_{1}} - {{{\kappa\beta cos}\left( {2\theta} \right)}{\overset{\_}{a}}_{1}}} \right\rbrack}}} & (7)\end{matrix}$where k₀=2π/λ_(c)=ω₀/c and a₁, ā₁, a₂, ā₂ stand for the field componentscorresponding to ±P_(1,2). For ā₁ and ā₂, analogous expressions may bewritten that differ from Eqs. (6) and (7) only in the signs of the sin θand cos θ terms (the signs follow the convention adopted for ±P_(1,2)).Terms proportional to κ_(i)≡m(β/2) {circumflex over (κ)}(G_(κi)) thatinvolve derivatives are neglected, since they are expected to be muchsmaller than the retained terms on the right-hand side. Similarequations hold for the TM polarization with the following substitutions:κ₁→−κ₁, κ₂ cos(2θ)→κ₂, κ₃ cos(2θ)→−κ₃.

The coefficients κ₁, κ₂, κ₃ correspond to the coupling from P₁ (30) to−P₁ (32), P₂ (34), and −P₂ (36), respectively, as well as the othercouplings that are equivalent by symmetry., In other words, κ₁ accountsfor DFB-like distributed reflection (by 180°), κ₂ represents the effectof diffraction by an angle 2θ from the “nearly-horizontal” Bragg planes,and κ₃ quantifies the diffraction by an angle (180°−2θ) from the“nearly-vertical” Bragg planes. For purely real κ (i.e. negligiblemodulation of the cavity loss by the grating), the following isobtained: $\begin{matrix}{\kappa_{i} = {\frac{2{\pi\Delta}\; n}{\lambda}\frac{1}{a_{L}}{\int_{R}^{\;}\mspace{7mu}{{\mathbb{d}^{2}x}\;{\exp\left( {{- {\mathbb{i}}}\;{G_{\kappa\; i} \cdot r}} \right)}}}}} & (8)\end{matrix}$where i=1, 2 or 3, Δn is the amplitude of the index modulation, R is thegrating feature (24) area, a_(L) is the area of the primitive cell ofthe reciprocal lattice, and G_(κ1)=|G₁−G⁻¹|=|G₂−G⁻²|,G_(κ2)=|G₁−G₂|=|G⁻¹−G⁻²|, G_(κ3)=|G₁−G⁻²|=|G⁻¹−G₂|. Equation (8) caneasily be generalized to the case of complex coupling by setting theimaginary part of Δn proportional to Δg or Δa_(i). For the circulargrating features (24) of radius r, shown in FIG. 1: $\begin{matrix}{\kappa_{i} = {\frac{2{\pi\Delta}\; n}{\lambda}\frac{\pi\; r^{2}}{\Lambda_{1}\Lambda_{2}}\frac{2{J_{1}\left( {G_{\kappa i}r} \right)}}{G_{\kappa i}r}}} & (9)\end{matrix}$

The optical gain g and the LEF α=−4π/λ (dn_(m)/dN)/(dg/dN) in {overscore(G)}=Γg(1−iα)−α_(i) are functions of the carrier density N at each pointin the laser plane. A simplifying approximation, that the diffusionlength is much smaller the typical scale of the variation of theenvelope function, is used here to calculate the carrier density. Thisapproximation is accurate in some cases of interest such as mid-IRquantum well lasers and will, in any case, not influence the mainfindings.Recombining all carriers at the same spatial position where they weregenerated by the electrical injection provides: $\begin{matrix}{\frac{\partial N}{\partial t} = \left\lbrack {\frac{j\left( {x,\mspace{11mu} z} \right)}{Med} - {c_{A}N^{3}} - {R_{sp}(N)} - {\frac{\Gamma\;{g(N)}}{\hat{h}\;\omega}\left( \left| a_{1} \middle| {}_{2}{+ \left| {\overset{\_}{a}}_{1} \middle| {}_{2}{+ \left| a_{2} \middle| {}_{2}{+ \left| {\overset{\_}{a}}_{2} \right|^{2}} \right.} \right.} \right. \right)}} \right\rbrack} & (10)\end{matrix}$where c_(A) is the Auger coefficient, R_(sp) is the spontaneous emissionrate, j is the injection current density, M is the number of periods(quantum wells) in the active region, and d is the thickness of oneperiod of the active region. The cross-terms that would appear in theparentheses are negligible owing to their fast spatial variation. Foroptical pumping, the first term in brackets should be replaced byI_(a)(Md

ω_(pump)), where I_(a) is the pump intensity absorbed in the activeregion and

ω_(pump) is the pump photon energy. The optical gain may be convenientlyparameterized in terms of differential gain and transparency carrierdensity, which in this design method are accurately calculated byincorporating complete energy dispersion relations and matrix elementsderived from a finite-element 8-band k·p algorithm or other bandstructure computation approaches. Any increase in the carrier or latticetemperature due to pumping of the active region is neglected, since thisis an acceptable approximation under many conditions of interest. Theinclusion of heating effects will alter only the quantitative detailsand will not influence the main design rules with regard to the photoniccrystal.

The design process can be simplified by making the assumption that thecenter wavelength has been chosen so that k₀n₀=mβ [i.e., the periodicityof the photonic lattice should be chosen so that Λ₂=mλ_(c)/(2n₀ cos θ)].A further assumption is that the angle θ (12) is relatively small.However, the aspect ratio (tan θ=Λ₂/Λ₁) cannot be too small, since thetransverse coherence is lost in that case. On the other hand, forTE-polarized modes if the aspect ratio is too large (close to 45°), thecoupling coefficients decrease. In that case, the beam corresponding tothe P₁ direction (30) propagates at such a large angle to the gainstripe (26) that much of the light is lost from the edges of the stripe.For a given laser structure, the novel theoretical formalism outlinedbelow will allow the determination of the optimal angle θ (12) thatyields the best balance between efficiency and beam quality.

Another assumption is that the spatial variation of the envelope isrelatively slow, i.e., mβ cos${\theta\frac{\partial a}{\partial z}}\operatorname{>>}{\frac{\partial^{2}a}{\partial z^{2}}\;.}$With these assumptions, four coupled time-dependent traveling-waveequations that are linear in both the time derivative and thepropagation-axis spatial derivative are obtained.

In the configuration shown in FIG. 1, the facets (18 and 20) are alignedwith one of the axes. In order to make the treatment of the boundaryconditions consistent with this, the following rotation by the angle θ(12) is applied: $\begin{matrix}{x = {\frac{x^{\prime}}{\cos\;\theta} + {z\mspace{11mu}\tan\mspace{11mu}\theta}}} & (11)\end{matrix}$thereby aligning a facet with the x′ axis. In performing thistransformation, second-order partial derivatives that tend to be smallare discarded, since they represent a slow variation. In spite of thegreat complexity presented by a 3-dimensional (2 spatial and 1 time)problem, the judicious simplifications introduced above make ittractable when solved by a split-step prescription that is known to beuseful in the time-domain modeling of 1D DFB lasers. This is describedin “An efficient split-step time-domain dynamic modeling of DFB/DBRlaser diodes”, B.-S Kim et al., IEEE J. Quantum Electron. Vol. (2000)and “Dynamic analysis of radiation and side-mode suppression in asecond-order DFB laser using time-domain traveling-wave model”, L. M.Zhang et al., IEEE J. Quantum Electron. Vol. (1994), which areincorporated herein by reference. A further simplification is effectedby using a beam-propagation-like fast Fourier transform (FFT) algorithmthat has been employed in the precise treatment of diffraction andcoupling effects in α-DFB lasers. This is described in“Fast-Fourier-transform based beam-propagation model for stripe-geometrysemiconductor lasers: Inclusion of axial effects”, G. P. Agrawal, J.Appl. Phys., Vol. (1984), which is incorporated herein by reference.

The novel time-dependent algorithm of the present invention will now bedescribed, using the rectangular lattice shown in FIG. 1. A uniformrectangular mesh with separations Δx′ and Δz can be chosen such that thedesign remains stable and convergent. In the absence of spontaneousemission, laser oscillation will not commence if the starting fieldintensity is zero. The spontaneous emission noise amplitude is generatedusing a pseudorandom number generator with a Gaussian random-variabledistribution. The noise satisfies the following correlation:$\begin{matrix}{{\left\langle {{s\left( {x^{\prime},\mspace{11mu} z,\mspace{11mu} t} \right)}{s^{*}\left( {x^{\prime},\mspace{11mu} z^{\prime},\mspace{11mu} t^{\prime}} \right)}} \right\rangle = {\hslash\;\omega_{0}\beta_{sp}M\;{R_{sp}\left( {x^{\prime},\mspace{11mu} z,\mspace{11mu} t} \right)}\frac{n_{0}}{c}{\delta\left( {t - t^{\prime}} \right)}{\delta\left( {z - z^{\prime}} \right)}}}\mspace{11mu}} & (12)\end{matrix}$The fraction of the spontaneous emission that is coupled into the modesof the optical field is taken to be β_(sp)=10⁻⁴. Assume that the noiseterms are the same for all field components.

The inventive design method also incorporates the spectral roll-off ofthe optical gain using a series expansion in powers of (ω−ω₀) [up to thequadratic term], where ω₀ is the gain spectrum's peak frequency, whichcan then be transformed into the time domain: $\begin{matrix}\left. {{g\left( {\hslash\;\omega} \right)} \approx {g_{0}\left\lbrack {1 - \left( \frac{{\hslash\;\omega} - {\hslash\;\omega_{0}}}{\Delta\;{\hslash\omega}_{g}} \right)^{2}} \right\rbrack}}\Rightarrow{{g(t)} \approx {g_{0}\left\lbrack {1 + {\frac{1}{\left( {\Delta\omega}_{g} \right)^{2}}\frac{\partial^{2}}{\partial t^{2}}}} \right\rbrack}} \right. & (13)\end{matrix}$Here g₀ is the peak gain and Δ

ω_(g) is the parameterized width of the gain spectrum as defined in Eq.(13). This expression is widely used since even multi-mode lasingusually occurs only close to the gain peak. Using an approximateconversion of the time derivatives into spatial derivatives, thefollowing equation is obtained: $\begin{matrix}\left. {\frac{1}{\left( {\Delta\omega}_{g} \right)^{2}}\frac{\partial^{2}a}{\partial t^{2}}}\Rightarrow{\left( \frac{n_{0}}{c\;\Delta\;\omega_{g}} \right)^{2}\frac{{{a\left( {{x^{\prime},\mspace{11mu} z} + {\Delta\; z,\mspace{11mu} t}} \right)} - {2{a\left( {x^{\prime},\mspace{11mu} z,\mspace{11mu} t} \right)}} + {a\left( {{x^{\prime}{,\;}\; z} - {\Delta\; z,\mspace{11mu} t}} \right)}}\;}{\left( {\Delta\; z} \right)^{2}}} \right. & (14)\end{matrix}$Following the split-step prescription of Kim et al., and applying it totwo-dimensional PCDFB structures, the effect of gain and carrier-inducedindex variations on the counter-propagating field components at time tis computed to provide the characteristic matrix, M_(dc) as follows:$\begin{matrix}{\begin{bmatrix}{a_{1}\left( {{x^{\prime},\mspace{11mu} z} + {\Delta\; z,\mspace{11mu} t}} \right)} \\{{\overset{\_}{a}}_{1}\left( {x^{\prime},\mspace{11mu} z\left. {,\; t} \right)} \right.} \\{a_{2}\left( {{x^{\prime},\mspace{11mu} z} + {\Delta\; z,\mspace{11mu} t}} \right)} \\{{\overset{\_}{a}}_{2}\left( {x^{\prime},\mspace{11mu} z,\mspace{11mu} t} \right)}\end{bmatrix} = {\begin{bmatrix}{\mathbb{e}}^{\hat{G}\;\Delta\; z\;\cos\;{\theta/2}} & 0 & 0 & 0 \\0 & {\mathbb{e}}^{\hat{G}\;\Delta\; z\;\cos\;{\theta/2}} & 0 & 0 \\0 & 0 & {\mathbb{e}}^{\hat{G}\;\Delta\; z\;\cos\;{\theta/{\lbrack{2{\cos{({2\theta})}}}\rbrack}}} & 0 \\0 & 0 & 0 & {\mathbb{e}}^{\hat{G}\;\Delta\; z\;\cos\;{\theta/{\lbrack{2{\cos{({2\theta})}}}\rbrack}}}\end{bmatrix}\begin{bmatrix}{{a_{1}\left( {{x^{\prime},\mspace{11mu} z,\mspace{11mu} t} - {\Delta t}} \right)}\;} \\{{\overset{\_}{a}}_{1}\left( {{x^{\prime},\mspace{11mu} z} + {\Delta\; z,\mspace{11mu} t} - {\Delta\; t}} \right)} \\{{a_{2}\left( {{x^{\prime},\mspace{11mu} z,\; t} - {\Delta\; t}} \right)}\;} \\{{\overset{\_}{a}}_{2}\left( {x^{\prime},\mspace{11mu} z\left. {{{+ \Delta}\; z,\mspace{11mu} t} - {\Delta\; t}} \right)}\; \right.}\end{bmatrix}}} & (15)\end{matrix}$where Δt=Δz(n₀/c).

Disregarding the gain terms, perform a Fourier transformation of thefield components with respect to the lateral spatial coordinateã(μ,z,t)=∫dμa(x′,z,t)exp(iμx′)  (16)and rewrite the coupled equations in the following form: $\begin{matrix}{{{\pm \frac{\partial\overset{\sim}{a}}{\partial z}} + {\frac{\cos\mspace{11mu}\theta\; n_{0}}{c}\frac{\partial\overset{\sim}{a}}{\partial t}}} = {{{iM}_{dc}\overset{\sim}{a}} = {{i\begin{bmatrix}{{- \frac{\mu^{2}}{2\beta\; m\;\cos\;\theta}} - {{\mu sin}\;\theta}} & {- \kappa_{1}^{\prime}} & {- \kappa_{2}^{\prime}} & {- \kappa_{3}^{\prime}} \\{- \kappa_{1}^{\prime}} & {\frac{\mu^{2}}{2\beta\; m\;\cos\;\theta} + {{\mu sin}\;\theta}} & {- \kappa_{3}^{\prime}} & {- \kappa_{2}^{\prime}} \\{- \frac{\kappa_{2}^{\prime}}{\cos\left( {2\theta} \right)}} & {- \frac{\kappa_{3}^{\prime}}{\cos\left( {2\theta} \right)}} & \frac{{{{- \mu^{2}}/\left( {2\beta\; m\;\cos\;\theta} \right)} + {\mu sin\theta}}\mspace{11mu}}{\cos\left( {2\theta} \right)} & {- \frac{\kappa_{1}^{\prime}}{\cos\left( {2\theta} \right)}} \\{- \frac{\kappa_{3}^{\prime}}{\cos\left( {2\theta} \right)}} & {- \frac{\kappa_{2}^{\prime}}{\cos\left( {2\theta} \right)}} & {- \frac{\kappa_{1}^{\prime}}{\cos\left( {2\theta} \right)}} & \frac{{{- \mu^{2}}/\left( {2\beta\; m\;\cos\;\theta} \right)} - {\mu sin\theta}}{\cos\left( {2\theta} \right)}\end{bmatrix}}\begin{bmatrix}{{\overset{\sim}{a}}_{1}\left( {\mu\left. {,\mspace{11mu} z,\; t} \right)} \right.} \\{{{\overset{\simeq}{a}}_{1}\left( {\mu,\mspace{11mu} z,\mspace{11mu} t} \right)}\;} \\{{\overset{\sim}{a}}_{2}\left( {\mu,\mspace{11mu} z,\mspace{11mu} t} \right)} \\{{\overset{\simeq}{a}}_{2}\left( {\mu,\mspace{11mu} z,\mspace{11mu} t} \right)}\end{bmatrix}}}} & (17)\end{matrix}$where M_(dc) is the characteristic matrix, κ₁′≡−κ₁ cos θ, κ₂′≡κ₂cos(2θ)cos θ, and κ₃′=−κ₃ cos(2θ)cos θ for TE-polarized light. ForTM-polarized light, the factors of cos(2θ) should be omitted in thedefinitions of κ₂′ and κ₃′, and the minus signs are omitted in thedefinitions of κ₁′ and κ₃′. In the case of complex coupling, complexconjugates of the appropriate coefficients should be employed in thelower triangular part of the matrix. In writing Eq. 17, ignore theslightly different factors of proportionality in front of the timederivative for different field components in order to make the problemmore tractable. The error becomes large only when θ is close to 45°, acase that is not important to the PCDFB. In the limit θ→0°, and forTE-polarized light as θ→45° (square lattice): κ₂′≈κ₃′→0. Furthermore, ifthe Λ₂ periodicity is removed, the α-DFB case is recovered with twooscillating waves coupled via κ₂′ (κ₁′=κ₃′=0). In that case, the matrixin Eq. 17 can be split into two 2×2 sub-blocks, which represent completedecoupling of the forward- and backward-propagating beams. On the otherhand, if θ=0° (or Δn=0), only the diagonal terms survive. Solving fortwo counter-propagating components, the algorithm presented here is thenequally applicable to the problem of gain-guided (broad-area)semiconductor lasers without a corrugation.

In the second stage of the split-step algorithm, analytically integrate${\pm \frac{\partial{\overset{\sim}{a}\left( {\mu,\mspace{11mu} t} \right)}}{\partial z}} = {{{iM}_{dc}{\overset{\sim}{a}\left( {\mu,\mspace{11mu} t} \right)}}:}$$\begin{matrix}{\begin{bmatrix}{\overset{\sim}{a}\left( {{\mu,\mspace{11mu} z} + {\Delta\; z,\mspace{11mu} t}} \right)} \\{\overset{\simeq}{a}\left( {\mu,\mspace{11mu} z,\mspace{11mu} t} \right)}\end{bmatrix} = {X_{dc}{\exp\left( {\mathbb{i}\Lambda}_{dc} \right)}{X_{dc}^{- 1}\begin{bmatrix}{\overset{\sim}{a}\left( {\mu,\mspace{11mu} z,\mspace{11mu} t} \right)} \\{\overset{\simeq}{a}\left( {{\mu,\mspace{11mu} z} + {\Delta\; z,\mspace{11mu} t}} \right)}\end{bmatrix}}}} & (18)\end{matrix}$where Λ_(dc) is a diagonal matrix containing the eigenvalues of the“diffraction-coupling” characteristic matrix M_(dc) as entries, andX_(dc) is a matrix containing the eigenfunctions of M_(dc). Thisoperation is preceded and followed by FFT transformations. For real Δn,the procedure expressed in Equation 18 is unitary, i.e., it conservesthe total power in the cavity. The operations in Equations 15 and 18 areperformed for each time step Δt.

Equations 15 and 18 embody the prescription for propagation of theoptical field. At each spatial position in the cavity at time t,numerically integrate Equation 10 simultaneously with the fieldpropagation, which provides values for the gain and LEF consistent withthe optical field. For the preferred embodiment, the gain stripe (26) isaligned or nearly aligned with the z axis. When applying boundaryconditions, it is a good approximation to neglect the reflected a₂components, which enter the cavity at large angles to the originaldirection:a ₁(x′,L)=√{square root over (R)}ā ₁(x′,L _(c)); ā ₁(x′,0)=√{square rootover (R)}a ₁(x′,0); a ₂(x′,L _(c))=0; a ₂(x′,0)=0.  (19)Therefore, at the facets (18 and 20), the a₂ component is taken to bezero.

The power emitted from a given facet (22) is determined by integratingthe near-field profile a₂(z=L_(c)) or ā₂ (z=0) over the gain stripe(26):P(I _(a))=(1−R)∫dx′|a ₁(x′,z=0,L _(c))|²  (20)The far-field emission characteristic may be determined from thenear-field profile a₂(z=L_(c)) or ā₂ (z=0) using the Fraunhoferdiffraction theory.

The spectral properties of the laser output (22) may be obtained fromthe power spectrum of the optical field, which is a Fourier transform ofthe autocorrelation function Ψ: $\begin{matrix}{{\Psi\left( {\tau,\mspace{11mu} x^{\prime}} \right)} = {\frac{1}{T}{\int_{0}^{T}\mspace{7mu}{{\mathbb{d}{{ta}_{1}^{*}\left( {x^{\prime};{z = {0,\mspace{11mu} L_{c}}};{t + \tau}} \right)}}{a_{1}\left( {x^{\prime};{z = {0,\mspace{11mu} L_{c}}};t} \right)}}}}} & (21)\end{matrix}$Here the time T must be long enough to capture the relevant detail ofthe laser spectrum. The power spectrum should be averaged over the,lateral (x′) direction in order to sample the spectral properties of theentire emitting aperture. In order to obtain steady-state values for allof the derived quantities, switch on the PCDFB laser and then wait untilall of the turn-on transients have dissipated. For the results discussedhere, convergence with respect to the mesh pitch and the time window Thave been demonstrated to occur.

The LEF (α) is defined as the ratio between the carrier-inducedvariations of the real and imaginary parts of the linear opticalsusceptibility. While the ratio varies somewhat with carrier density, atlow temperatures and densities it may be considered constant. The plasmacontribution to the refractive index, which scales as the square of thewavelength, should be included into the calculation of the differentialindex in the numerator. The LEF is calculated at the lasing photonenergy, since it is a spectral function, and apart from the plasmacontribution vanishes at the peak of the differential gain. The mismatchbetween the peak gain, where lasing occurs barring strong spectralselectivity of the cavity loss, and the peak differential gain is theimmediate cause of a non-zero LEF in a semiconductor laser, and isitself a consequence of the asymmetric conduction and valence-banddensities of states in the active region. The LEF can be thought of as ameasure of the focusing (α<0) or defocusing (α>0) of the optical waveinduced by a positive carrier-density perturbation. In a region wherespatial hole burning leads to a slight local decrease of the carrierdensity, a positive LEF induces self-focusing, which is ultimatelyresponsible for filamentation. The product (Γg_(th)α) of the thresholdgain (Γg_(th)), which includes contributions due to the mirror loss,internal loss, and diffraction loss, and the LEF (α) is typically themost critical parameter that limits the beam quality and spectral purityof high-power single-mode lasers, including the PCDFB. The correctinclusion of the LEF into the treatment of the beam quality and spectralcharacteristics of PCDFB lasers is essential to performing a reliabledesign analysis.

The appropriate loss for a given laser and set of operating conditionsshould be determined self-consistently, since the internal loss candepend on the pumping conditions and the diffraction loss depends on themode propagation properties. While, in principle, both the internal lossand the diffraction loss depend in a complicated way on spatial positionwithin a given laser cavity, the variations tend to be relatively smalland in evaluating the optimal PCDFB parameters, it is adequate to employaverage values. Several of these different possibilities will beconsidered. The criterion for placing a given laser in one of thecategories will be the product of the threshold modal gain Γg_(th) andthe LEF α, where Γg_(th) is equal to the sum of the internal loss,facet-reflectivity loss, and diffraction loss associated with thegrating. The diffraction loss in a single-mode PCDFB laser can rangeroughly from 10 cm⁻¹ to 30 cm⁻¹ depending on the stripe width andemission wavelength.

The inventive method can be used to design lasers in a number ofdifferent classes. One class concerns lasers with a large Γg_(th)α(≧100cm⁻¹), examples of which include double-heterostructure or even somequantum-well interband lasers operating at high carrier densities orwith a strongly broadened gain spectrum, particularly in the mid-IR.Another class concerns lasers with intermediate values of the Γg_(th)αproduct (on the order of several tens of cm⁻¹⁾, an example of which isan InP-based quantum-well laser designed to operate in thetelecommunications wavelength range of 1.3–1.55 μm, in which no specialmeasures are taken to reduce the LEF). A third class includes laserswith small values of Γg_(th)α(≦20 cm⁻¹), examples of which arelow-loss/low-LEF GaAs-based lasers, quantum-dot (QD) lasers and quantumcascade lasers (QCLs). All of these device classes are typicallydesigned to emit TE-polarized light, with the exception of the QCL,which is limited solely to TM-polarized light emission.

The inventive design method provides a beam output for the selectedelements of rectangular lattices, where the highest single-mode powercan be obtained by using 16°≦θ≦23° for a wide range of Γg_(th)α. In alattice with square or circular grating features (24), it is difficultto obtain a value of κ₃′ that is much different from κ₁′, and thedifference between them has relatively little effect on the PCDFB laserperformance. For definiteness, unless otherwise specified, |κ₃′|=|κ₁′|(but note that the exact value of κ₃′ is not critical). Furthermore, thehighest single-mode power can be obtained for a cavity length L_(c) (28)of at least 1.5 mm (for most devices the upper limit may be about 2.5 mmdue to practical considerations) with only a weak dependence onΓg_(th)α. For a rectangular-lattice PCDFB, the maximum stripe width(W_(max)) which can maintain single-mode operation as a function of theΓg_(th)α product is shown in FIG. 2. This dependence is relativelyinsensitive to the wavelength and other details of a given laser,although in self-consistently determining the Γg_(th)α product it mustbe remembered that the diffraction loss scales approximately withwavelength.

Whereas various grating orders higher than first ({1,1} in the notationdefined above) may be used, the refractive index perturbation requiredto obtain the same coupling coefficients in the optimized range islarger for higher-order couplings, and the possible waveguide losssource due to out-of-plane emission must be considered. A furtherimportant consideration is the possibility of accidental degeneraciesbetween two resonances with different m and l. For example, when θ=tan⁻¹(7^(−1/2))≈20.7°, i.e., in the optimized grating angle (12) range, theresonance wavelengths for the {2,2} and {5,1} grating-order resonancesof the rectangular lattice coincide. A general relation for the gratingangle 0<θ≦45° for which the {l,m} and {l′,m′} grating orders are inresonance is as follows: $\begin{matrix}{\theta = {\tan^{- 1}\sqrt{\frac{m^{2} - m^{\prime 2}}{l^{\prime 2} - l^{2}}}}} & (22)\end{matrix}$Possible near-degeneracies are to be avoided if stable operation in asingle mode is desired. On the other hand, in some cases it may bedesirable to intentionally cause the lasing wavelength to controllablyhop from one grating resonance to another by means of tuning the peak ofthe gain spectrum with temperature or pump intensity/current density.When the gain spectrum is positioned in between two resonances,simultaneous highly coherent lasing at two separated wavelengthscorresponding to the resonances can be obtained. Thisdiscrete-tuning/two-color laser approach based on two nearly degenerateresonances of the rectangular lattice also falls within the scope ofthis invention.

In the case of lasers with a large LEF (≧100 cm⁻¹), a hexagonal latticeconfers no advantage, because at such large Γg_(th)α the coherence islost at relatively narrow stripe widths, and the decrease in efficiencymore than offsets the improvement in the beam quality. A rectangularlattice with 16°≦θ≦23° and |κ₃′|≈|κ₁′| is therefore optimal. Theinventive method demonstrates that the maximum single-mode power can beobtained when 1.5≦|κ₁′|L_(c)≦2.5 (or 8 cm⁻¹ ≦|κ₁′|≦15 cm⁻¹ in thepreferred embodiment with 1.5 mm≦<L_(c)≦2.5 mm). The results of theoptimized design that are shown in FIG. 2 imply that for largeLEF/Γg_(th)α, the maximum stripe widths are ≦150 μm, and the bestoperation is obtained for 150 cm⁻¹ ≦|κ₂′|≦250 cm⁻¹ (which corresponds to2.0≦|κ₂′|W≦2.5). Thus the PCDFB optimization mandates that relativelylarge coupling coefficients be used when Γg_(th)α is large. For m=1,these results imply that the optimum side of the square grating feature(24) is 0.9–0.95 Λ₂, while for m=2, a≈0.55 Λ₂. For the example of anoptically pumped mid-IR “W” laser with L_(c)=2 mm and λ=4.6 μm, themaximum stripe (26) width for which single-mode output can be obtainedis≈100 μm. The external quantum efficiency is then 55% of that for aFabry-Perot laser with the same parameters, while the beam quality andspectral purity are far superior.

For the range 40 cm⁻¹≦Γg_(th)α≦100 cm⁻¹, the optimal |κ₁′|L_(c) falls inthe range 1–2 (corresponding to 4 cm⁻¹≦|κ₁′|≦15 cm⁻¹), the maximumstripe (26) width is 150–350 μm, as seen from FIG. 2, and the otimal κ₂′is in the range 60–150 cm⁻¹ (1.5≦|κ₂′|W≦2.5). For m=1, these resultsimply that the optimum side of the square is 0.85–0.9 Λ₂, while for m=2,a≈0.57–0.58 Λ₂.

When Γg_(th)α is lower, the required inter-feature spacing decreasessomewhat. For a rectangular lattice with Γg_(th)α≦40 cm⁻¹, the TDFTsimulations yield optimal results for |κ₃′|≈|κ₁′|, 2 cm⁻¹≦|κ₁′|≦10 cm⁻¹(0.5≦|κ₁′|L_(c)≦1.5), 350 μm≦W_(max)≦450 μm, and 30 cm⁻¹≦|κ₂′|≦60 cm⁻¹(1.3≦|κ₂′|W≦2.1). For m=1, these results imply that the optimum side ofthe square is 0.8–0.85 Λ₂, while for m=2, a≈0.6 Λ₂.

The preceding configuration specifications are based on the objective ofmaximizing the single-mode output power by maximizing the stripe (26)width. However, these specifications should be modified if good (but notnecessarily diffraction-limited) multi-mode beam quality is a moreimportant objective than spectral purity. To achieve that objective thecoupling coefficients should be reduced somewhat. For Γg_(th)α≧100 cm⁻¹,1.5≦|κ₁′|L_(c)≦2 (6 cm⁻¹≦|κ₁′|≦13 cm⁻¹) is optimal, while1≦|κ₁′|L_(c)≦1.5 (4 cm⁻¹≦|κ₁′|≦10 cm⁻¹) is optimal for 40cm⁻¹≦Γg_(th)α≦100 cm⁻¹, and 0.5≦|κ₁′|L_(c)≦1.5 (2 cm⁻¹≦|κ₁′|≦10 cm⁻¹) isoptimal for Γg_(th)α≦40 cm⁻¹. For all Γg_(th)α, it is optimal to employ0.5≦|κ₂′|W≦1.5 when the best beam quality is the overriding objective.

The present invention is not limited to designing the PCDFB laser typewith a rectangular lattice geometry (10). Other laser types andgeometries are included without departing from the scope of the presentinvention. For example, a hexagonal geometry (40) may be used for thephotonic crystal instead of the rectangular lattice geometry (10)discussed as part of the preferred embodiment. This configuration isshown in FIG. 3. Instead of two equivalent propagation directions, thereare now three [P₁=(0, 1) (56); P₂=(√3/2, 1/2) (58); and P₃=(−√3/2, 1/2)(60)], so that six distinct components of the optical field must beconsidered. In addition, the gain stripe (50) is aligned with P₁ (56),and the laser facets (44 and 46) are perpendicular to P₁ (56). Thusapart from the pattern of the grating features (42), thehexagonal-lattice PCDFB laser is somewhat simpler because the gainstripe (50) is not tilted with respect to the facets (44 and 46).However, in the absence of coatings, the hexagonal geometry (40) isgenerally less attractive for lasers with a large LEF. This is due tomicroscopic refractive-index fluctuations which will favor parasiticFabry-Perot-like modes that propagate between the facets (44 and 46)without being significantly affected by the narrow grating resonance,thereby robbing power from the desired resonant mode and degrading theoverall beam quality. Moreover, in the hexagonal geometry (40),transverse coherence is established by beams that propagate at ±60° toP₁ (56) This means that diffraction losses will be high unless the gainstripe (50) is relatively wide. Using the inventive design method for again medium with a small LEF-threshold gain product, such as a quantumcascade laser employing intersubband transitions or a low-lossGaAs-based laser, the hexagonal-lattice PCDFB (40) produces a lasingmode with the highest degree of optical coherence when the pump stripe(50) is very wide. For gain media with a small LEF-threshold gainproduct, it will therefore be advantageous over all other geometries,including the rectangular-lattice PCDFB (10), when the objective is tomaximize the output laser beam (48) power that can be emitted into asingle optical mode with high spectral purity andnear-diffraction-limited beam quality. This has not been proposedpreviously for edge-emitting semiconductor lasers.

Distinct hierarchies of coupling orders are obtained, which depend onthe orientation direction. Considering first propagation along the Γ-Jdirection in reciprocal space (see FIG. 3), the first two orders withcoupling at the J point [β=(4/3)π/Λ and β=(8/3)π/Λ], where Λ is theperiod (68) of the hexagonal lattice (40), allow coupling between onlythree directions at angles of 120° to each other (e.g. P₁ (56), P⁻² (64)and P⁻³ (66)) due to Condition 2 (which was described earlier). On theother hand, the third and fourth J-point orders admit coupling betweenall six equivalent directions. For coupling occurring at the X point ofthe reciprocal lattice, there are also two kinds of diffractiveorders: 1) with two opposite propagating directions only [e.g.,β=(2√3/3)π(2l+1)/Λ, l=0, 1, 2, . . . ] and 2) with four equivalentpropagation directions (here, the coupling resembles that in arectangular lattice with θ=30°). For the six-fold coupling that takesplace at the Γ point with propagation vectors pointing along the Γ-Xdirections, with k=β(0,0), where β=(4√3/3)mπ/Λ and Λ is the period ofthe hexagonal lattice [Λ=(2√3/3)mλ_(c)/n₀, m=1, 2, 3, . . . ], therelevant reciprocal lattice vectors are: G₁=2β(0,1), G⁻¹=2β(0,−1),G₂=2β(√3/2,1/2), G⁻²=2β(−√3/2,−1/2), G₃=2β(−√3/2,1/2), andG⁻³=2β(√3/2,−1/2).

Returning to the Γ-X orientation, there are three distinct couplingcoefficients, which can be calculated from an expression similar toequation (3). A convention is adopted whereby κ₁ corresponds to couplingstraight back (e.g., 180° rotation, from P₁ to P⁻¹, etc.), κ₂ is thecoefficient responsible for diffraction by a 60° angle (e.g., from P₁ toP₂, P₁ to P₃, or P⁻² to P₃), and κ₃ is the coefficient responsible fordiffraction by a 120° angle (e.g., from P₁ to P⁻², P₁ to P⁻³, or P₂ toP⁻¹). In order to implement the inventive time-dependent Fouriertransform (TDFT) approach, an approximation,${{\beta\frac{\partial a}{\partial z}}\operatorname{>>}\frac{\partial^{2}a}{\partial z^{2}}},$is employed. Note that neglecting the second derivative in z is anessential feature of all TDFT calculations. Fortunately, for cases wherethe spatial coherence of the optical beam is relatively strong, thisapproximation is expected to be very good. The results of the analysisfor the hexagonal lattice (40) can then be written in terms of thecharacteristic matrix M_(dc), defined as in Equation 17 except that thecos θ term on the left-hand side is omitted: $\begin{matrix}{M_{d\; c} = \begin{bmatrix}{{- \mu^{2}}/\left( {2\;\beta} \right)} & {- \kappa_{1}} & {- \kappa_{2}} & {- \kappa_{3}} & {- \kappa_{2}} & {- \kappa_{3}} \\{- \kappa_{1}} & {{- \mu^{2}}/\left( {2\;\beta} \right)} & {- \kappa_{3}} & {- \kappa_{2}} & {- \kappa_{3}} & {- \kappa_{2}} \\{{- 2}\;\kappa_{2}} & {{- 2}\;\kappa_{3}} & {{{- \mu^{2}}/\beta} - {\mu\sqrt{3}}} & {{- 2}\;\kappa_{1}} & {{- 2}\;\kappa_{3}} & {{- 2}\;\kappa_{2}} \\{{- 2}\;\kappa_{3}} & {{- 2}\;\kappa_{2}} & {{- 2}\;\kappa_{1}} & {{{- \mu^{2}}/\beta} + {\mu\sqrt{3}}} & {{- 2}\;\kappa_{2}} & {{- 2}\;\kappa_{3}} \\{{- 2}\;\kappa_{2}} & {{- 2}\;\kappa_{3}} & {{- 2}\;\kappa_{3}} & {{- 2}\;\kappa_{2}} & {{{- \mu^{2}}/\beta} + {\mu\sqrt{3}}} & {{- 2}\;\kappa_{1}} \\{{- 2}\;\kappa_{3}} & {{- 2}\;\kappa_{2}} & {{- 2}\;\kappa_{2}} & {{- 2}\;\kappa_{3}} & {{- 2}\;\kappa_{1}} & {{{- \mu^{2}}/\beta} - {\mu\sqrt{3}}}\end{bmatrix}} & (23)\end{matrix}$Equation (23) is applicable to TM-polarized light with ã≡└ã₁ {overscore(ã)}₁ ã₂ {overscore (ã)}₂ ã₃ {overscore (ã)}₃┘, whereas the sameexpression with coefficients κ₂ and κ₃ reduced by a factor of 2 andκ₁→−κ₁, κ₃→−κ₃ holds for TE-polarized light. Although inaccuracies inthe treatment of the time derivatives for the P₂ (58) and P₃ (60) beamsbecome somewhat larger here than in the rectangular-lattice case, thatcan be at least partly mitigated by employing a finer mesh. Thetreatment of the time variation for the P₁ beam (56), which is thesource of the normal emission from the facets, is exact.

For Γg_(th)α≦20 cm⁻¹, higher single-mode output powers and better beamqualities are attainable using the hexagonal lattice (40) of FIG. 3rather than a rectangular lattice (10) such as that in FIG. 1. Toachieve the maximum single-mode power, κ₁L_(c)≈1 should be employed.However, in the Γg_(th)α≦20 cm⁻¹ region, the hexagonal lattice (40) ispreferable to a rectangular lattice (10) when the stripe width (50)becomes wider than about 800 μm, although quite high single-mode outputpowers can be obtained with narrower stripes (50). The maximum stripe(50) width for which single-mode operation is expected is about 1.5 mm,for coupling coefficients in the range 20 cm⁻¹≦|κ₂|, κ₃|≦60 cm⁻¹. The κ₂and κ₃ coefficients are nearly equal in the preferred embodiment andlarger than κ₁ by at least a factor of 3. Since the period Λ (68) of thehexagonal lattice (40) is fixed by the desired resonance wavelength λ,the index-modulation-amplitude is the only other variable parameter inthe lateral extent of the grating features (42). Circular gratingfeatures (42) are preferred in the case of a hexagonal lattice (40),since when the grating feature (42) shape departs from a circle, thecoupling between different waves becomes non-equivalent. More than threecoupling coefficients would be required to describe the operation of thedevice, and the performance would be expected to suffer to some extent(although the hexagonal lattice (40) may still be advantageous over arectangular lattice (10) if for practical reasons it is more convenientto fabricate grating features (42) that are not strictly circular).Fortunately, a wide variety of different κ sets can be realized by usingcircles alone, fr which the radius is related to the filling fractionvia F=(2π/√3)r²/Λ². The |κ₂|≈|κ₃|>κ₁ condition can also be realizedusing the lowest-order six-fold coupling at the Γ point with thediameter of the circular region being about 90–97% of the length of theperiod, i.e., the adjacent grating features (42) are nearly touching.The performance of the device is relatively insensitive to variations inthe circle diameter of no more than a few percent. The optimizedconfigurations range from |κ₁′|=2 cm⁻¹, |κ₂′|=|κ₃′|=7 cm⁻¹, L_(c)=1 mmfor W=2.8 mm to |κ₁′|=2 cm⁻¹, |κ₂′|=|κ₃′|=25 cm⁻¹, L_(c)=480 μm forW=0.8 mm to |κ₁′|=10 cm⁻¹, |κ₂′|=|κ₃′=120 cm⁻¹, L_(c)=300 μm for W=100μm (we set a limit of L_(c)=300 μm on the minimum cavity length). Thesecoupling coefficients are realized by using the lowest-order resonancealong the Γ-X direction and varying the fill factor of circular gratingfeatures (42). Considerably smaller cavity (54) lengths (≈1 mm) arerequired to achieve the best beam quality in hexagonal-lattice devices(40) for the following reasons: (1) the gain stripe (50) is not tilted,so a longer cavity (54) less effectively discriminates againstFabry-Perot-like modes owing to the smaller associated threshold gain ofthose modes; and (2) the P₂ (58) and P₃ (60) beams propagate at ratherlarge angles of ±60° with respect to the cavity (54) axis, which impliesthat a large aspect ratio between the stripe (50) width and the cavity(54) length is expected to yield the maximum brightness.

One version of the PCDFB laser with the parameters slightly outside therange of optimal values has been reduced to practice. A piece ofantimonide laser material with an InAs/GaInSb/InAs/AlAsSb “W”quantum-well active region and GaSb top separate-confinement region waspatterned by optical lithography into a second-order rectangular-latticephotonic-crystal configuration (10) in accordance with the main approachof the invention but with grating parameters falling outside therequired ranges due to limitations of the available optical lithography.The grating was second-order to allow larger grating features (24) andpermit patterning with optical lithography, although better optimizedfirst-order gratings may be created by electron-beam lithography.Reactive ion etching into the top GaSb layer produced 90-nm-deepcircular grating features (24) on a rectangular lattice (10). Followingthinning of the substrate, laser facets were cleaved to a cavity lengthof about 2 mm. The photonic crystal was oriented at an angle of 20°relative to the facet normal, in accordance with the invention. Thedevice was mounted epitaxial-side-up using thermal compound, and opticalpumping was provided by 100-ns pulses from a Ho:YAG laser. The lasingwavelength was λ=4.6 μm, and the operating temperature at which the gainspectrum matched the photonic-crystal resonance was 183 K.

Although the spectrum was not single mode, because the grating did nothave optimized parameters and also because the pulsed excitation inducedchirp associated with carrier and lattice heating, the spectrum wasnonetheless a factor of 4–10 narrower than that of a Fabry-Perot lasermade from the same wafer, and also narrower than earlier “W” lasersemitting in the same wavelength region. Proper working of the PCDFBdevice was confirmed because the peak wavelength varied only weakly withtemperature, which was expected since the wavelength is determinedprimarily by the photonic-crystal resonance condition rather than by thepeak wavelength of the gain spectrum. On the other hand, the outputwavelength for an earlier α-DFB laser made from the same material variedrapidly with temperature, following the gain peak, and also displayedlittle spectral narrowing. It should be noted that there was also a neardegeneracy of the {2,2} and {5,1} grating-order resonances that wasdiscussed above in connection with Eq. (22). This led to thesimultaneous observation of both resonant wavelengths at someintermediate temperatures.

The beam quality, as quantified in terms of the etendue (basically, theproduct of the output beam's divergence angle and its effective aperturesize) was also substantially improved over the earlier α-DFB device. Theetendue for the PCDFB laser was smaller at all pump-stripe (26) widths,and the difference was as large as a factor of 5 at stripe widths of 200μm and above. This represents a substantial improvement over anunpatterned Fabry-Perot geometry. These tests of a non-optimized versionof the invention confirm that the PCDFB laser operates in the intendedmode, and also that it displays narrower spectral linewidth and improvedbeam quality over the prior available designs. Further details are givenin a journal article published by the inventors with experimentalcollaborators, W. W. Bewley, C. L. Felix, I. Vurgaftman, R. E. Bartolo,J. R. Lindle, J. R. Meyer, H. Lee, and R. U. Martinelli, “Mid-infraredphotonic-crystal distributed-feedback lasers with improved spectralpurity and beam quality”, Appl. Phys. Lett., vol. 79, pp. 3221–3223(2001)), which is incorporated herein by reference.

The present invention includes lasers built using the specificationproduced by this design method. However, the present invention alsoincludes lasers constructed with a particular grating configuration, andwhose operation is made possible or is enhanced by the presence of thePCDFB grating. One of the unique configurations made possible byphotonic crystals is an edge-emitting highly coherent broad-area laser(70) that requires only a single facet (80). An example of thisconfiguration is shown in FIG. 4. A hexagonal-lattice PCDFB laser may bedefined in the manner outlined in connection with FIG. 3. However,instead of pumping the entire area between two specially defined laserfacets, only a roughly triangular (74) or roughly semicircular (76) areain the vicinity of the one facet (80) is pumped. In the preferredembodiment, the single required facet (80) is AR-coated. The product ofthe coupling coefficients and the radius (or the full spatial extent) ofthe pumped area (74 or 76) is kept at a reasonably large value thatallows most of the light to be diffracted before reaching the boundaryof the pumped region (74 or 76). Therefore, the numerical value for theproduct of any coupling coefficient and the full spatial extent of thepumped region should be no smaller than 2. The device shown in FIG. 4operates by 2D feedback from the hexagonal grating. Some of the lightoscillating inside the pumped area (74 or 76) is extracted from thefacet (80). The emitted power is controlled by varying the spatialextent of the pumped area (74 or 76) and the ratio of the facet (80)surface to the circumference of the pumped area (74 or 76).

The high degree of spatial coherence provided by the hexagonal latticemay also be utilized for the purposes of beam steering as shown in FIG.5 (90). In order to realize the beam-steering capability, one of thefacets (92) of the hexagonal (FIG. 3) or tilted rectangular (FIG. 1)PCDFB lasers should be curved rather than planar. The desired degree ofcurvature may be realized by fabricating the facet (92) by etchingrather than the cleaving technique routinely employed for edge-emittingsemiconductor lasers. In the preferred embodiment, an AR coating isapplied to the curved facet (which is assumed to have a constant radiusof curvature) in FIG. 5, whereas an HR coating is applied to the flatfacet (96) in the same figure. The gain stripes (93, 94, and 96 areexamples) may then be located at various positions with respect to thecurved facet (92). The moment of the radius of the curvature is muchgreater than the stripe (93, 94, or 96) width, so that the facet (92)curvature may be neglected within a single stripe (93, 94, or 96). Ifthe stripe (94) is positioned to be perpendicular to the tangent to thecurved facet (92), then the output beam (102) for that stripe willemerge at nearly normal incidence to the tangent. On the other hand, ifthe tangent makes some small angle φ_(i) with the normal to the axis ofthe stripe (93 and 96 are examples), the output beam (100 and 104 arethe corresponding output beams) will also be emitted at a non-zero angleφ_(e) with respect to the normal to the tangent. The emission angleφ_(e) is amplified with respect to φ_(i) by the modal refractive indexof the laser waveguide, in accordance with Snell's law. It is possibleto maintain lasing in the stripes at an angle with the normal to thetangent to the curved facet (92) surface, since the distributed feedbackeliminates the need for facet reflections in the PCDFB structure. Thisallows beams from individual stripes (93, 94, and 96) to be steered overa relatively large range of angles with the ultimate constraints on itsbounds imposed by the collection efficiency. The gain stripes (93, 94,and 96) may be defined by electrical contacts or by optical pumping. Inthe former case, no special fabrication procedure apart from thedefinition of the photonic crystal and the curved facet (92) isnecessary, and it is possible to use a single pump beam to obtainemission at a wide range of angles.

In FIG. 6, the configuration (l10) includes a radius of curvature of thefacet (112) that is comparable to the stripe (116) width, with thesingle stripe (116) having a variable width. In the preferredconfiguration, the stripe (116) width is varied by optical pump beamswith different pump spots. The stripe (116) is assumed to beperpendicular to the tangent of the curved facet (112), so that when thestripe (116) is relatively narrow, the output beam (118) will emergenearly at normal incidence as discussed in the previous paragraph. Asthe stripe (116) width is increased, the different parts of the coherentwavefront of the lasing beam (118) near the curved facet (116) will bediffracted into free space outside the facet at different angles asrequired by Snell's law. Owing to the strong lateral coherence, the neteffect is that the emitted beam converges towards a focal point (120) atsome distance away from the curved facet (112). This distance isdetermined by the relation between the facet (112) curvature and thestripe (116) width in a unique manner. Beyond the focal point (120), thebeam (118) diverges while maintaining a beam quality close to thediffraction limit. This ability to vary the angle of convergence and thefocal point is important for a number of practical applications.

Another configuration (130) builds on the microdisk laser conceptfamiliar to those skilled in the art. Feedback in a microdisk laser isprovided by glancing reflections from the circular lateral boundary ofthe device's active region. If the disk diameter is too large (on theorder of a few hundred microns), most of the area of the disk has littleoverlap with the lasing mode, which is localized near the boundary.Furthermore, the extraction of the light from the microdisk laser isproblematic, insofar as only a small fraction of the oscillating lightis capable of escaping from the disk via photon tunneling. FIG. 7 showsa microdisk laser (130) that can effectively utilize the entire activearea when its diameter is as large as several hundred microns. Themicrodisk laser of FIG. 7 allows a drastic enhancement in the poweremitted into a single mode at the expense of defining a photonic-crystalgrating (132) throughout the active area. The square lattice ispreferred for the TM polarization. For the TE polarization, a hexagonallattice with six equivalent directions should be used. Inside themicrodisk, lasing proceeds in a manner that is similar to the otherPCDFB configurations, and strong spatial and spectral coherence ismaintained via multiple distributed-feedback processes as discussed inconnection with the rectangular-lattice and hexagonal-lattice PCDFBlasers. Insofar as only a single emission direction is required forlight extraction, in some embodiments the output power is maximized byreflecting the beams that would have emerged in the other three (fivefor the hexagonal lattice) directions. This may be accomplished bymetallizing approximately three quarters (five sixths) of the activedevice area with the coating (136) oriented with respect to the gratingas shown in FIG. 7, by incorporating Bragg gratings at the other threeemission directions, or by other means. An AR coating may be applied tothe remaining quarter of the device boundary (134).

The inventive grating configuration allows tunability to be built intothe laser. Three new approaches to the tunability problem are presented:(1) multi-section wide-stripe tunable lasers; (2) photonic crystallasers that are tunable by varying the angle that the lasing modepropagating along the cavity axis makes with the crystal lattice; and(3) PCDFB lasers with optically-tunable gratings. Within each of theseapproaches, there are a variety of alternative configurations.

Referring to FIG. 8, a 2D photonic crystal with laser cavity is shownwith facets (146 and 148) and a gain stripe bounded by the facets (146and 148) and two edges (142 and 144). The laser cavity is divided intothree sections: a first (PCDBR) tuning section (156 at the right of thefigure), a central section (150), and a second tuning section (158, atthe left of the figure). In both tuning sections (156 and 158), themodal refractive index is tunable via the applied voltage as will bediscussed further below. Two or three separate electrodes (not shown)are used to inject carriers into these different cavity sections (150,156, and 158). Although the following discussion assumes electricalinjection, all of the configurations discussed here could alternativelyoperate by using two or three independently controllable optical pumpingbeams instead. In that case, a very precise alignment of the pump beamswould be required owing to secondary pumping of the active material bylasing photons. Facets (146 and 148) appear at each end of the photoniccrystal. The facets (146 and 148) of the device in FIG. 8 may be definedeither by cleaving or by lithography, e.g., using reactive ion etching.

A central section (150) of the device comprises a first 1D grating withperiod (Λ₁, 152) that is defined by standard methods. The grating isinclined at some angle (θ₂, 154) with respect to the facets, but isperpendicular to the boundaries of the central section (162 and 164). Ina preferred embodiment, the angle θ₂ (154) will be close to 20°. In thefirst tuning section at the right side of FIG. 8, an oblique-lattice 2DPCDBR mirror is defined in a preferred embodiment. The first grating(166) is continued into the first (right) tuning section (156) at thesame orientation as in the central section (150). The oblique 2D latticecomprises a superimposed second grating (168), having period (Λ₂, 160)and an orientation that is parallel to the facets (146 and 148). In apreferred embodiment, the left boundary (164) of the first (right)tuning section (156) is perpendicular to the first grating (166) in thecentral region (150), while the left boundary (169) of the second (left)grating (168) is parallel to the facets (146 and 148). As is illustratedby FIG. 8, this leaves a triangular region that is covered by the righttuning electrode (not shown) but does not contain the second (2D)grating. In the preferred embodiment, an antireflection (AR) coating isapplied to the right facet (148), and a high-reflectivity (HR) coatingis applied to the left facet (146). However, the choice can also be madenot to use coatings. In the preferred embodiment, the device isgain-guided, with all of the electrodes oriented so as to create a(nearly) continuous pump stripe of width W parallel to the first grating(166).

As photons are generated in the central section (150) by stimulatedemission induced by application of the voltage V₁, they undergo Braggreflections at the angle θ₁ shown in the figure. This occurs when theirwavelength is close to the grating resonance given by the relation:λ₁=2Λ₁ n₁sin θ₁, where n₁ is the modal refractive index determined bythe laser waveguide with confinement along the growth direction as wellas by the injected carrier density in the central section (150). In thefirst tuning section (156, at the right of FIG. 8), the refractive indexn₂ is controlled by varying the voltage, V₂, applied to that section.The resonance wavelength in the first tuning section (156) is λ₂=2Λ₁n₂sin φ. This is due to the fact that one of the superposed gratings inthe first (right) tuning section is the first grating (166) (from thecentral section(150)) and therefore has the same period (152). In thefirst tuning section (156), the propagation angle φ must be equal toangle θ₂ (154), the angle by which the grating is tilted with respect tofacet (148) since the AR coating on the right facet (148) prevents anysignificant feedback from facet (148). Therefore, the necessarycondition for assuring that feedback from the second (2D) grating (168)(at wavelength λ₂=2 Λ₂ n₂) returns along the original path ofpropagation (at an angle θ₂ (154) with respect to the 1D grating (166))is λ₂=2Λ₁n₂ sin θ₂. These two expressions fix the relation between thetwo periods (152 and 160) as Λ₂=Λ₁ sin θ₂. Feedback occurs almostentirely along the original path as long as the length of the firsttuning section (156) is shorter than the exchange length of the 1Dgrating (166) in the central section (150). Use Snell's law (n₁ sinθ₁=n₂ sin θ₂) at the interface between the two sections (150 and 156) toobtain the angle θ₁ in terms of angle θ₂ (a fixed structural parameter),n₁ (also fixed under any given set of operating conditions), and n₂(which is controllable via V₂). Clearly the resonance wavelengths in thetwo sections (150 and 156) are identical (λ₁=λ₂), and the cavity modeclosest to the resonance will experience the least loss, whereas othermodes will be suppressed. The consequence is that the lasing wavelengthmay be controlled by V₂. In some cases it will be possible to tune thePCDBR resonant wavelength over the entire gain spectrum, e.g., by asmuch as±5%.

To tune the operating wavelength in the structure of FIG. 8, n₁ andangle θ₂ remain nominally fixed, whereas n₂, angle θ₁ and λ are altered.In fact, angle θ₁ and λ are uniquely determined by the tuned-in value ofn₂. The grating period Λ₁ (152) is selected so as to optimize theoperation of the central region (150). For example, if the average valueof the modal index in the tuning section (156) <n₂> is assumed to be n₁:Λ₁=<λ>/(2 n₁ sin θ₂), the relation between the two periods (152 and 160)may be expressed as Λ₂≡Λ₁ sin θ2=<λ>/(2 n₁). The use of a tilted gratingin the central and first tuning sections allows the laser beam tomaintain transverse coherence in that section. If only the secondgrating (168) with period Λ₂ (160) is utilized, the maximum single-modestripe width would be reduced considerably, and the result would be amuch lower single-mode output power.

In addition to the central section (150) and the first (right) tuningsection (156), the preferred embodiment also includes a second tuningsection (158) (shown to the left of central section in FIG. 8). Thesecond tuning section (158) is oriented at angle θ₂ (154) with respectto the HR-coated facet (146). The presence of the second tuning section(154) is desirable to insure that the laser beam strikes the HR-coatedfacet (146) at normal incidence. Since it is desired that the refractiveindices in the first and second tuning sections (156 and 158) be thesame (n₂), the applied voltages should be similar. However, a 1D grating(166) may be employed in the second tuning section (158), rather than a2D grating (166 and 168) as is used in the first tuning section (156).It is also possible to use the second tuning section (158) for phasecontrol by fine-tuning the refractive index in that section to a valuethat is slightly different from n₂. An alternative is to eliminate thesecond tuning section (158) and to rely on self-correction of the angleat the left facet (146).

This inventive configuration allows the output to emerge normal to acleaved facet (148) irrespective of the wavelength, and also maintainscoherence for quite wide gain-guided stripes. As a result, theattainable output powers are much higher than for any earlierwavelength-tunable single-mode semiconductor laser.

Some specific design requirements for the tunable PCDBR laser design ofFIG. 8 follow. (1) The length (L₁) of the central section (150) may bechosen either to produce quasi-continuous tuning (in which case it willbe long) or to enhance the differential efficiency (in which case itwill be shorter), with preferred values being between 0.5 mm and 2.5 mm.(2) The coupling coefficient (κ_(a)) of the first (central) 1D grating(166), is determined by optimizing the high-power single-mode outputcorresponding to the width W of gain stripe. (3) The length (L₂) of the2D grating in the first tuning section is then selected so that it issmaller than the exchange length associated with the first grating(166). In practice, L₂ should usually be in the 50–100 μm range sincethe exchange length is roughly given by 1/(2κ_(a)). (4) The lateralwidth of the first tuning section (156) is chosen so that the gainstripe completely overlaps the second grating (168). (5) The couplingcoefficient, κ_(b), of the second grating (168) is selected so as tomaximize the efficiency of the device while maintaining sufficientlystrong wavelength discrimination (e.g., κ_(b)L₂ of about 1 yields aneffective reflectivity of approximately 50%). In most cases, κ_(a) andκ_(b) will be comparable and fall in the 50–300 cm⁻¹ range. This is incontrast to PCDFB devices, in which κ₁′ and κ₃′ are usually about anorder of magnitude smaller than κ2′, and typically fall in the 5–20 cm⁻¹range.

A number of means exist for varying the refractive index, n₂, in thePCDBR section (156) over a large enough range to provide substantialwavelength tuning. The most straightforward method is to use the voltageapplied to the PCDBR electrode (not shown) to modify the carrier densityinjected into the section (156). The maximum achievable carrier densitywill be limited by leakage (e.g., thermionic emission over the electronbarriers), amplified spontaneous emission, and other mechanisms. Thepublication “Optimization of the carrier-induced effective-index changein InGaAsP Waveguides—Application to tunable Bragg filters”, J. P.Weber, IEEE J. Quantum Electron. (1994). yielded an upper limit of3–4×10¹⁸ cm⁻³ for InGaAsP-based devices operating at telecommunicationswavelengths (1.3–1.55 μm). Preferably the energy gap in the activelayer(s) of the tuning region (156) is large compared to the lasingphoton energy, so that the first tuning region (156) is essentiallytransparent to the laser radiation. This is necessary to avoidequalization of the carrier densities in the central section (150) andthe tuning section (156) by stimulated emission that would occur if thetwo energy gaps were roughly the same. In other embodiments, the indextuning rate may be enhanced by adding a contribution due to interbandabsorption in the first tuning section (156). For telecommunicationslasers, appropriate InGaAsP-based layer structures have already beenformulated and tested experimentally, although other materialcombinations would also be suitable. Different band gaps can be realizedin sections by selective-area epitaxy, by other epitaxial regrowthtechniques, or by etching away the small-gap layers in the tuningsections (156 and 158). Two InGaAsP active regions with slightlydifferent compositions can be close in total thickness as well as in therefractive index, and therefore create only a small mismatch between thewaveguide mode in the central region (150) and its continuation into thetuning section (156). Doping of the tuning layer (156) can also help toreduce the mode mismatch between the two sections (150 and 156).

For antimonide mid-IR lasers with higher electron barriers, the maximumcarrier density will be on the order of the highest achievable dopingdensity of nearly 10¹⁹ cm⁻³. The index tuning is enhanced significantlyin longer-wavelength devices, since the plasma contribution to therefractive index increases as λ². The rate of relative wavelength tuningdue to the plasma index shift near the center wavelength λ is given bythe following expression: $\begin{matrix}{\frac{\Delta\;\lambda}{\lambda} = {{- 0.0165}\left( {1 + \frac{f_{h}}{f_{e}}} \right)\;{\Gamma_{SCH}\left( \frac{\Delta\; N}{10^{18}\mspace{11mu}{cm}^{- 3}} \right)}\left( \frac{3.3}{n_{1}} \right)^{2}\left( \frac{0.04}{m^{*}} \right)\left( \frac{\lambda}{4\mspace{20mu} µ\; m} \right)^{2}}} & (24)\end{matrix}$where ƒ_(h)/ƒ_(e), is the ratio of the electron-to-hole contributions tothe refractive index, expected to be on the order of 10% even for equaldensities of injected electrons and holes due to the much heavier holemass, ΔN is the difference in the injected carrier densities in the twosections, which for mid-IR devices with high electron barriers canpotentially be as high as 10¹⁹ cm ⁻³, and m* is the electron effectivemass in the separate confinement heterostructure (SCH) layer into whichthe carriers are injected. Thus, for mid-IR devices the plasmacontribution alone can shift the wavelength by nearly 400 nm, or 10%, ofthe wavelength when λ=4 μm. If desired, the gain bandwidth can beartificially enhanced by employing multiple quantum wells withintentionally chirped well widths. The expected 10% variation in therefractive index corresponds to an angular modulation of angle θ₁ byonly about 2°, which will make the PCDBR tunable lasers quite robust.Wavelength-tunable antimonide mid-IR PCDBR structures have been designedin detail, and are found to combine large tuning ranges with high outputpowers.

The tuning which corresponds to a given injected carrier density may insome cases be enhanced by additionally making use of interband andintervalence absorption contributions to the refractive index. This isaccomplished by positioning the energy gap or the split-off gap in theseparate-confinement region of the laser waveguide sufficiently close tothe photon energy, but not so close that the extra absorption reducesthe efficiency of the device excessively.

The wavelength-tunable designs specified above do not necessarilyrequire that a photonic crystal be used in the first tuning region(156). The main advantage of the PCDBR is simply that it allows opticalcoherence to be maintained over a much wider gain stripe, and thereforeenables higher powers to be emitted into a single mode.

Many configurations other than the preceding multi-sectionwavelength-tunable PCDBR lasers (which used variability of therefractive index in the PCDBR tuning section) are possible. Theseinclude separable-grating schemes such as the sampled-grating techniquein which a second DBR grating (168) is periodically interrupted byregions with zero coupling coefficients. The two DBR mirrors withslightly mismatched periods and a separate phase section produceextensive wavelength tuning by aligning distinct resonances in the twomirrors. This approach is particularly adaptable to the incorporation ofthe inventive PCDBR mirrors in place of the usual DBR mirrors, becausethe grating periods remain uniform (or nearly so). Other schemes basedon superstructure gratings or chirped gratings can also be adapted, butpresent a greater cost in complexity since the periods of the first andsecond gratings must be varied simultaneously. In all of these cases,the maximum output power of the wavelength-tunable single mode issubstantially enhanced when inventive PCDBR mirrors are employed becausecoherence can be maintained for a much wider stripe.

The fact that the output emerges along the normal to a facet,irrespective of the operating wavelength can be critical in applicationswhere the laser light must be coupled into a fiber or other opticalsystem, and where beam steering cannot be tolerated. The tunable PCDBRconcept shares with multi-section DBR devices the possibility ofbroad-range tuning and is compatible with other proposed enhancements ofthe tuning range via modification of the second grating in FIG. 8.Furthermore, it is expected to outperform related configurations, inwhich either the first 1D or the 2D PCDBR section of FIG. 8 is omitted.In the former case, the maximum attainable single-mode power would belower, since transverse coherence would be poorer in the PCDBR section.The latter case would represent a multi-section α-DFB structure. Themulti-section α-DFB laser is less attractive than the multi-sectionPCDBR, since the cleaved facets in the α-DFB case do not providewavelength-sensitive feedback. The tuning mechanism would then berequired to rely on the accuracy to which the angle θ₂ would remainfixed as n₂ is varied. The PCDBR scheme is considerably more robust,will provide a larger side-mode suppression ratio, and will not requirethat the Γg_(th)α product be very small. This is due to novel featuresof the PCDBR design including the tilt of the section interfaces withrespect to the facets, the oblique-lattice superposition of gratings(166 and 168) in the tuning section (156), and the prospects for etchingaway the active region in the tuning section (156) in order to take fulladvantage of the plasma index shift.

The mirror (146) beyond the second tuning section (left side of FIG. 8),which is preferably HR, can be defined by cleaving or etching (at thesame or a different angle from the AR-coated mirror adjacent to thePCDBR mirror). It can be a coated or uncoated facet, or reflection maybe provided by a DBR or PCDBR region. The active (150) and PCDBR (156)sections can be defined by a single epitaxial growth with post-growthprocessing such as etching, or alternatively by selective-area epitaxyor another regrowth strategy. Tuning sections (156 and 158) with a widevariety of energy gaps relative to the active region fall under thescope of the invention. The energy gap in the tuning section can be muchlarger than that in the active section, just slightly larger, or thesame as in the active section (not preferred). The modal refractiveindex in the tuning sections (156 and 158) can be modulated by manypossible mechanisms, including plasma shift and modulation of theinterband, intervalence, intersubband, or free or bound exciton orimpurity-level absorption. Absorption modulation could involveengineering of the conduction or valence band edge density of states, orabsorption processes involving states far away from the band edges.

The two-dimensional nature of the feedback in photonic-crystal lasersprovides an additional inventive mechanism for varying the wavelengththrough angle tuning, which takes advantage of the fact that in aphotonic crystal the resonance wavelength is not fixed by the period asin DFB and DBR lasers, but varies as a function of the propagationdirection. For example, if the propagation direction is by some meansaligned with the short period Λ₂ of the photonic crystal geometryconsidered in connection with FIG. 1, any variation of the propagationdirection by some angle φ will also produce wavelength tuning by afactor of 1/cos φ. A similar effect may be obtained in an obliquelattice as in FIG. 9 (220) defined by a 2D periodic array of features(226), in which there are no two degenerate axes (such as (222) and224)), which leads to more choices of possible orientation of thepropagation direction. Note that this configuration relies on a singlepropagation direction with a variable angle, rather than on thepropagation along two coupled off-axis directions that occurs inwide-stripe PCDFB lasers.

Embodiments employing the angle tuning mechanism combine the 2D natureof the photonic crystal, a well-defined propagation direction that isnot determined by the photonic crystal properties, single-modeoperation, and a convenient means of switching between different angles(wavelengths). A well-defined propagation direction can be selected byusing a ridge waveguide, which must be narrow enough (at most severaltens of photonic crystal periods) to insure single-mode operation. Inthat case the periodic structure outside of the ridge will beinterrupted, and only relatively coarse wavelength multiplexing can berealized. Nevertheless, spanning the wavelength range in relativelylarge increments using an array of waveguides oriented at differentangles with respect to the common photonic crystal grating may be usefulin some applications. For example, discrete wavelengths at δλ intervalscorresponding to roughly 1–2% of the central wavelength λ₀ will beattainable, so that a typical gain bandwidth Δλ/λ of 10% would produce atotal of 5–10 discrete wavelengths. A larger total number of wavelengthsis attainable if the gain bandwidth can be broadened, artificially orotherwise. The definition of the facets is inessential in thisconfiguration, insofar as feedback is provided in a distributed mannerby the grating. It is estimated that the minimum increment in thisarrangement will be on the order of one percent of the centerwavelength, even in a structure with a small Γg_(th)α product. Theoutput of the discrete laser waveguides can be combined using amultimode interference (MMI) optical combiner. The waveguide sectionsnext to the combiner may be gently curved in order to assure that thebeam exiting the MMI combiner couples efficiently to an opticalamplifier or another optical system. A schematic of a possible deviceconfiguration (170) employing the angle tuning mechanism is shown inFIG. 10, where only 5 waveguides (176, 177, 179, 181, and 183) are shownrepresenting 5 wavelengths. The five wavelengths are located in a commonPCDFB region (182) and are connected via an MMI combiner (172) to anamplifier (174). This arrangement is confined between a first facet(180) having an HR coating and a second facet having an AR coating(184). Either electrical or optical pumping may be employed to excitethe individually contacted waveguides (176, 177, 179, 181, and 183),which may be turned on independently or in combination.

The extra step of defining individual ridge waveguides (176, 177, 179,181, and 183) is not strictly necessary if optical pumping is employed.By tailoring the pump beam to vary the angle (178) it makes with thephotonic-crystal structure of the sample, gain-guided devices withpump-angle-based wavelength tuning may be realized. The advantage ofthis configuration is a reduced complexity of the fabrication process.

The mechanism of wavelength tuning by changing the propagation direction(propagation angle) in a photonic crystal has the advantage of astraightforward selection of different wavelengths from photonic crystaldevices monolithically fabricated on a single wafer. To accomplish asimilar wavelength multiplexing with conventional DFB lasers, thegrating pitch in each device must be carefully adjusted in eachwavelength selection. The fabrication procedure for a gain-guidedangle-tunable photonic crystal laser is particularly straightforward,since no ridges need to be defined.

A third approach to wavelength tuning by angle variation based on thePCDBR configuration (190) is shown schematically in FIG. 11. Awell-defined propagation direction at some angle (196) with respect to anormal to a facet (192) is selected via an array of single-mode ridgewaveguides (194, 202, 204, 206, and 208) in the otherwise-unpatternedsection of the sample. The optical output is taken out of one end of thesingle-mode waveguides (194, 202, 204, 206, and 208) and is terminatedby an AR-coated facet (192). The waveguides (194, 202, 204, 206, and208) can be curved near the facet (192) in order to obtain output beamsthat are nearly normal to the facet, which also removes the limit onmaximum internal angle that can be coupled out of the device. At theother end, the waveguides are slowly broadened in a tapered section(200), which is gradual enough to preserve their single-mode nature. Thetapered ends are fed into a common broad-area PCDBR mirror (198). ThePCDBR mirror (198) provides feedback at a wavelength that varies withthe direction of propagation (depending on which individual waveguide isexcited). The tapering in this case is required to take full advantageof the 2D nature of the photonic crystal in the PCDBR mirror.

When rectangular photonic crystals effectively consist of thesuperposition of two distinct gratings, a vanishing value of theDFB-like reflection coefficient κ₁′ is produced (whereas the other twocoupling coefficients can retain large values corresponding to Braggreflections from each of the gratings). Since the gratings areseparable, they do not necessarily need to co-exist in the same layer.In fact, the physical nature of the two gratings can be quite differentas long as the net index modulation is adjusted to be comparable, andthe resulting-coupling coefficients are in a favorable range. Theinventive method uses separable gain-coupled gratings based onpatterning with an optical pump.

In one embodiment, an optical pump may be used to pattern the photoniccrystal even in two dimensions, e.g., by employing a 2D microlens arrayor 2D optical interference. If carrier diffusion is neglected, then theinjected carrier density will replicate the pattern of the pump beam.Modulation of the carrier density will affect both the refractive indexand the optical gain, although gain coupling is expected to dominate.However, the Λ₂ period of an optimized first-order PCDFB structure isless than a micron even at mid-IR wavelengths, whereas typical diffusionlengths in mid-IR antimonide laser materials tend to be close to amicron at cryogenic temperatures and decrease to a few tenths of amicron at room temperature. Therefore, it is unlikely that a fulloptically generated 2D PCDFB pattern can survive unless higher-ordermid-IR or long-wavelength IR lasers are operated at elevatedtemperatures.

This carrier-diffusion limitation can be effectively circumvented inanother embodiment (200) shown in FIG. 12 by physically defining one ofthe gratings (210) by etching, while creating the other grating (208) byspatially modulating the pump beam in the perpendicular dimension. Theetched grating (210) can operate either by index coupling or gaincoupling. Since an optically defined first-order grating (208) designedfor mid-IR wavelengths will have a period of several microns, it islikely to survive even in the presence of appreciable carrier diffusionat low temperatures. The concept can be extended to the near-IR spectralregion by employing higher-order optically defined gratings. In order toobtain a 1D grating from a uniform pump beam, a number of opticalsystems can be utilized. One of these includes positioning a 0/−1 phasemask (a configuration known to those skilled in the art) close to thesample surface, creating an interference pattern with a period of a fewmicrons at the sample surface. Optical interference between the twohalves of a split pump beam to form a transient grating may also beemployed. The concept can further be extended to electrically pumpedsystems, in which a grating can be created along or both axes by theselective epitaxial deposition of a p-n junction between the contactsand the active region.

When a patterned optical pumping beam is used to induce one of thegratings (208), the lasing wavelength can be tuned by changing theperiod of that grating. This can be accomplished by tilting the 0/−1phase mask with respect to the sample surface, varying the optical pathlength between the phase mask and the sample surface, varying theincidence angle for the transient grating, etc. It should be noted thatthe wavelength modulation induced by the spatially-modulated opticalpumping will be accompanied by beam steering, corresponding to aninternal angle varying as the inverse tangent of the ratio between thetwo periods.

One advantage of the photonic-crystal scheme with one etched and oneoptically induced grating is that there is no need to transfer a 2Dpattern to the sample. The grating definition could be equivalent to theprocedures commonly employed in conventional 1D DFB lasers, such asholographic patterning or optical lithography. The operating wavelengthis continuously tunable by varying the period of the optical grating.

The inventive laser design provides diffraction in three differentdirections, including DFB-like reflection, that are inherent to thePCDFB laser operation. The results for the beam quality of the emittedlight are defined in terms of the etendue. Etendue is the product of thestandard deviations of the near-field pattern and the far-field pattern,the former measuring the spatial extent of the emitted beam at the facetand the latter quantifying the angular divergence of the diffracted beamaway from the facet. It is well known in the art that the minimumetendue for a Gaussian beam, referred to as the diffraction limit, isproportional to the operating wavelength. FIG. 13 illustrates theresults from the inventive calculation for the etendue as a function ofstripe width for the case of LEF=4, a 2-mm-long cavity, and pumping by atop-hat profile at 14 times threshold (I=14I_(th)). Etendues are shownfor the cases of Fabry-Perot (FP), α-DFB, and rectangular-lattice PCDFBlasers. The coupling coefficient for the α-DFB laser is 200 cm⁻¹, andthe coupling coefficients for the PCDFB device are κ₁′=κ₃′=10 cm⁻¹ andκ₂′=200 cm⁻¹, with θ=20°. Whereas the FP laser has quite poor beamquality for all of the considered stripe widths (≧50 μm), both the α-DFBand PCDFB outputs are near-diffraction-limited for widths narrower than≈100 μm. As the pump stripe is broadened, however, the etendue for thePCDFB laser increases much more slowly, remaining below 10 times thediffraction limit (DL) for stripes as wide as 500 μm. This resultdemonstrates the considerable advantages of the inventive PCDFB with theoptimized parameter space outlined above as compared to the prior stateof the art (FP and α-DFB).

FIG. 14 shows calculated etendues for FP, α-DFB, and rectangular-latticePCDFB lasers identical to those in FIG. 13, except that LEF=2. The PCDFBperforms better than the α-DFB over the whole range of stripe widths,and both significantly outperform the FP laser. The slight increase inthe etendue at narrow stripe widths is due to the smaller κ₂′=50 cm⁻¹employed in order to assure the best performance at wider stripe widths.The range of parameters illustrated in FIG. 14 should be mostadvantageous for some classes of quantum-well lasers such as near-IRunstrained devices, telecommunications lasers, and good-quality mid-IR“W” lasers. We note that while in the case of telecommunications laserseither a smaller internal loss or a smaller LEF (e.g., by p-typemodulation doping) can in principle be obtained, in practice it hasremained challenging to demonstrate a small Γg_(th)α product.

FIG. 15 shows the etendues for FP, α-DFB, and rectangular-lattice PCDFBlasers with LEF=0.2, the value estimated for a typical QCL emitting atλ=4.6 μm. The non-zero value of the LEF arises from a combination of theconduction-band nonparabolicity and the plasma contribution to therefractive index. The latter becomes more pronounced at longerwavelengths, whereas the former is rarely expected to exceedapproximately 0.1 for the InGaAs/AlInAs and GaAs/AlGaAs material systemscurrently used to fabricate QCLs. The results in FIG. 15 may also beapplicable to high-quality interband near-IR lasers in which theinternal loss is no more than 5 cm⁻¹, whereas the LEF can be on theorder of or less than 1, and to quantum-dot lasers, for which very lowLEF values (0.1–0.5) have been measured. Although the beam qualityprovided by the α-DFB in FIG. 13 is also good, the PCDFB laser yieldsnearly diffraction-limited output (with a slight deviation of theetendue due to the slightly non-Gaussian near field) for stripes as wideas 1.5 mm. The robustness of the single-mode spectral purity of therectangular-lattice PCDFB is superior to that of the α-DFB, and alsoexpect further improvements at the widest stripewidths when ahexagonal-lattice PCDFB is employed. The FP structure once again showsmuch poorer beam quality.

The PCDFB approach of the present invention also provides a substantialextension of the conditions under which single-mode operation can besustained in the presence of LEF-driven index fluctuations. This isshown in FIG. 16 which illustrates emission spectra for 100-μm-widePCDFB and α-DFB lasers with Γg_(th)α=40 cm⁻¹. Whereas the PCDFB devicehas a side-mode suppression ratio (SMSR) of >>30 dB, the α-DFB output isclearly multi-mode. FIG. 2 shows a contour plot, as a function ofΓg_(th)α, of the stripe widths at which the PCDFB (light gray), α-DFB(gray) and conventional DFB (black) lasers operate in a single mode. Inorder to readily discriminate against noise without resorting to a veryfine mesh and/or unreasonably long run time, a rather lax condition,SMSR≧20 dB, is used to define single-mode output. The PCDFB is seen tohave a far more extensive single-mode region than either the α-DFB orDFB devices. This implies that PCDFB single-mode output powers may be upto an order of magnitude higher. When the LEF is very small, as oftenoccurs in QCLs, QD lasers, and specially-optimized quantum-well lasers,it is advantageous to employ a hexagonal lattice. Whereas the maximumoutput powers for the rectangular-lattice PCDFB is only somewhat largerthan for the α-DFB laser in that limit, the optimized hexagonal-latticePCDFB lattice produces a single-mode output at stripe widths of up toapproximately 1.5 mm, and yields single-mode powers that are higher thanfor the α-DFB by a factor of approximately 2.5.

Even in the cases of single-mode operation for both α-DFB and PCDFBlasers, the spectral purity in the PCDFB is much more robust owing tothe near absence of side modes. The suppressed yet still-observable sidemodes in the α-DFB laser usually become much more pronounced underpulsed operating conditions (owing to transient heating effects, etc.).Therefore, whereas α-DFB lasers have proven to be relativelyunattractive single-mode sources for applications requiring fast directmodulation, optimized electrically-pumped PCDFB devices will operate upto the maximum modulation frequencies characteristic of state-of-the-artconventional DFB devices, i.e., in the GHz range. The maximum modulationfrequencies of optically-pumped PCDFB lasers are substantially higher,up to 100 GHz, owing to the elimination of parasitic RC time constants.Of course, in the latter case the pump laser light must be modulated atthe same frequency. Finally, the results presented in FIGS. 2 and 13–16are attainable only in the optimized PCDFB configuration of the presentinvention with emission normal to the facet. Designs that do not employparameters in the range specified in this invention may yieldperformance that is no better or even worse than that of α-DFB devices.

The PCDFB and PCDBR concepts disclosed here are broadly applicable tovirtually any in-plane semiconductor laser emitting at any wavelengthfrom the ultraviolet to the terahertz range in any material system orpolarization of light.

PCDFB and PCDBR lasers can be pumped either electrically or optically,they can employ index-coupled, gain-coupled, or complex-coupledgratings, and they may have coated or uncoated facets defined bycleaving, etching or other technologies known to those skilled in theart. The gratings may incorporate one-dimensional or two-dimensional λ/4phase shifts in order to improve the stability and reproducibility ofthe single-mode operation at the grating resonance. The gain stripes maybe tapered in order to increase the single-mode power levels, usingmethods analogous to flared lasers and amplifiers known to those skilledin the art. The gratings in the PCDFB lasers can be chirped (linearly orquadratically) in order to suppress spatial-hole burning near the centerof the laser waveguide at high output powers. For rectangular-latticePCDFB lasers, one or the other period can be chirped independently, orboth periods can be chirped simultaneously. The chirped PCDFB lasers mayreach higher single-mode output powers, by analogy with chirpedconventional DFB lasers.

In addition to rectangular photonic crystal lattices, the inventionincludes PCDFB lasers having a wide range of values of the Γg_(th)αproduct. Hexagonal photonic crystal lattices may be used for wide-stripePCDFB lasers with small Γg_(th)α and other photonic crystal symmetrieswill display similar advantages. Slight misalignments from a truerectangular lattice or true hexagonal lattice are typically notdetrimental to the laser performance.

While square, rectangular, or circular grating features (24 or 42) areoften preferred under certain conditions as discussed, in general thePCDFB or PCDBR laser operation is not very sensitive to the exact shapeof the grating features (24 or 42) as long as they produce couplingcoefficients in the optimal ranges. In some designs it is not evenrequired that the grating features (24 or 42) be totally distinct. Forexample, in designs where the optimal spacing between grating features(24 or 42) is very small along one axis, a corrugated line may havesimilar properties to a series of nearly-touching squares, rectangles,or circles, as long as the coupling coefficients are made to fall in theoptimal ranges.

Combinations of the single-grating and PCDBR regions other than those ofthe preferred embodiment can also be used to tune the high-powersingle-mode output. The main criterion in such designs is the selectionof a single wavelength using the PCDBR mirror and the use of thesingle-grating section to maintain spatial coherence in the emittingsection. PCDBR mirrors with lattice symmetries other than oblique canalso be used in those configurations.

The invention has been described in what is considered to be the mostpractical and preferred embodiments. It is to be recognized, however,that obvious modifications to these embodiments may occur to thoseskilled in this art. Accordingly the scope of the invention is to bediscerned from reference to the appended claims.

1. A photonic-crystal distributed-feedback laser with an output power, a beam quality, a wavelength spectrum, and a product of a modal loss and a linewidth enhancement factor, comprising: a laser cavity with a waveguide structure that has a cavity length L_(c) and is bounded by two mirrors; an active region for producing optical gain upon receiving optical pumping or an input voltage; at least one layer including a periodic two-dimensional grating with modulation of a modal refractive index, said grating being defined on a rectangular lattice with a first period along a first axis of said grating and a second period along a second perpendicular axis of said grating; said grating produces three diffraction processes having coupling coefficients κ₁′, κ₂′, κ₃′; a lateral gain area contained within a second area patterned with said grating that has substantially a shape of a gain stripe with a width W, with said gain stripe tilted at a first tilt angle relative to the two mirrors; and wherein said rectangular lattice of said grating is tilted at a second tilt angle substantially the same as the first tilt angle with respect to said gain stripe and the ratio of said first and second grating periods is equal to the tangent of said first tilt angle, with said first tilt angle being between about 16° and about 23°.
 2. A photonic-crystal distributed-feedback laser as in claim 1, with said cavity length at least 1.5 mm.
 3. A photonic-crystal distributed-feedback laser as in claim 2, wherein said product of the modal loss and linewidth enhancement factor is no larger than about 40 cm⁻¹.
 4. A photonic-crystal distributed-feedback laser as in claim 3, wherein said stripe width W is between about 350 μm and about 450 μm and wherein said output power into a single spectral mode is optimized by requiring said coupling coefficients, said cavity length L_(c), and said stripe width to substantially satisfy the relations: |κ₃′|≈|κ₁′|; 2 cm⁻¹≦|κ₁′|≦10 cm⁻¹; 0.5≦|κ₁′|L_(c)≦1.5; 30 cm⁻¹≦|κ₂′|≦60 cm⁻¹; and 1.3≦|κ₂′|W≦2.1.
 5. A photonic-crystal distributed-feedback laser as in claim 3, wherein said output power and said beam quality are both optimized by requiring said coupling coefficients, said cavity length L_(c), and said stripe width W to substantially satisfy the relations: |κ₃′|≈|κ₁′|; 2 cm⁻¹≦|κ₁′|≦10 cm⁻¹; 0.5≦|κ₁′|L_(c)≦1.5; and 0.5≦|κ₂′|W>1.5.
 6. A photonic-crystal distributed-feedback laser as in claim 2, wherein said product of the modal loss and linewidth enhancement factor is between about 40 cm⁻¹ and about 100 cm⁻¹.
 7. A photonic-crystal distributed-feedback laser as in claim 6, wherein said stripe width W is between about 150 μm and about 350 μm and wherein said output power into a single spectral mode is optimized by requiring said coupling coefficients, said cavity length L_(c), and said stripe width to substantially satisfy the relations: |κ₃′|≈|κ₁′|; 4 cm^(−1≦|κ) ₁′|≦15 cm⁻¹; 1.0≦|κ₁′|L_(c)≦2.0; 60 cm⁻¹≦|κ₂′|≦150 cm⁻¹; and 1.5≦|κ₂′|W≦2.5.
 8. A photonic-crystal distributed-feedback laser as in claim 6, wherein said output power and said beam quality are both optimized by requiring said coupling coefficients, said cavity length L_(c), and said stripe width W to substantially satisfy the relations: |κ₃′|≈|κ₁′|; 4 cm⁻¹≦|κ₁′|≦10 cm⁻¹; 1≦|κ₁′|L_(c)≦1.5; and 0.5≦|κ₂′|W≦1.5.
 9. A photonic-crystal distributed-feedback laser as in claim 2, wherein said product of modal loss and linewidth enhancement factor is larger than about 100 cm⁻¹.
 10. A photonic-crystal distributed-feedback laser as in claim 9, wherein said stripe width W is no larger than about 150 μm and wherein said output power into a single spectral mode is optimized by requiring said coupling coefficients, said cavity length L_(c), and said stripe width to substantially satisfy the relations: |κ₃′|≈|κ₁′|; 8 cm⁻¹≦|κ₁′|≦15 cm⁻¹; 1.5≦|κ₁′|L_(c)≦2.5; 150 cm⁻¹≦|κ₂′|≦250 cm⁻¹; and 2.0≦|κ₂′|W≦2.5.
 11. A photonic-crystal distributed-feedback laser as in claim 9, wherein said output power and said beam quality are both optimized by requiring said coupling coefficients, said cavity length L_(c), and said stripe width W to substantially satisfy the relations: |κ₃′|≈|κ₁′|; 1.5≦|κ₁′|L_(c)≦2; and 1.5≦|κ₂′|W≦2.0. 